Under what conditions does correlation imply causation? We all know the mantra "correlation does not imply causation" which is drummed into all first year statistics students. There are some nice examples here to illustrate the idea.
But sometimes correlation does imply causation. The following example is taking from this Wikipedia page

For example, one could run an experiment on identical twins who were known to consistently get the same grades on their tests. One twin is sent to study for six hours while the other is sent to the amusement park. If their test scores suddenly diverged by a large degree, this would be strong evidence that studying (or going to the amusement park) had a causal effect on test scores. In this case, correlation between studying and test scores would almost certainly imply causation.

Are there other situations where correlation implies causation?
 A: *

*Almost always in randomized trials

*Almost always in observational study when someone measure all confouders (almost never)

*Sometimes when someone measure some counfounders (IC* algorithim of DAG discovery in Pearl's book Causality)

*In non gaussian linear models with two or more variables but not using correlation as measure of relationship (LiNGAM)


Most of discovery algorithms are implemented in Tetrad IV
A: One useful sufficient condition for some definitions of causation:
Causation can be claimed when one of the correlated variables can be controlled (we can directly set its value) and correlation is still present. 
A: A related question might be -- under what conditions can you reliably extract causal relations from data?
A 2008 NIPS workshop try to address that question empirically. One of the tasks was to infer the direction of causality from observations of pairs of variables where one variable was known to cause another, and the best method was able to correctly extract causal direction 80% of the time.
A: Correlation is not sufficient for causation. One can get around the Wikipedia example by imagining that those twins always cheated in their tests by having a device that gives them the answers. The twin that goes to the amusement park loses the device, hence the low grade.
A good way to get this stuff straight is to think of the structure of Bayesian network that may be generating the measured quantities, as done by Pearl in his book Causality. His basic point is to look for hidden variables. If there is a hidden variable that happens not to vary in the measured sample, then the correlation would not imply causation. Expose all hidden variables and you have causation.
A: I'll just add some additional comments about causality as viewed from an epidemiological perspective. Most of these arguments are taken from Practical Psychiatric Epidemiology, by Prince et al. (2003).
Causation, or causality interpretation, are by far the most difficult aspects of epidemiological research. Cohort and cross-sectional studies might both lead to confoundig effects for example. Quoting S. Menard (Longitudinal Research, Sage University Paper 76, 1991), H.B. Asher in Causal Modeling (Sage, 1976) initially proposed the following set of criteria to be fulfilled:


*

*The phenomena or variables in question must covary, as indicated for example by differences between experimental and control groups or by nonzero correlation between the two variables.

*The relationship must not be attributable to any other variable or set of variables, i.e., it must not be spurious, but must persist even when other variables are controlled, as indicated for example by successful randomization in an experimental design (no difference between experimental and control groups prior to treatment) or by a nonzero partial correlation between two variables with other variable held constant.

*The supposed cause must precede or be simultnaeous with the supposed effect in time, as indicated by the change in the cause occuring no later than the associated change in the effect.


While the first two criteria can easily be checked using a cross-sectional or time-ordered cross-sectional study, the latter can only be assessed with longitudinal data, except for biological or genetic characteristics for which temporal order can be assume without longitudinal data. Of course, the situation becomes more complex in case of a non-recursive causal relationship.
I also like the following illustration (Chapter 13, in the aforementioned reference) which summarizes the approach promulgated by Hill (1965) which includes 9 different criteria related to causation effect, as also cited by @James. The original article was indeed entitled "The environment and disease: association or causation?" (PDF version).

Finally, Chapter 2 of Rothman's most famous book, Modern Epidemiology (1998, Lippincott Williams & Wilkins, 2nd Edition), offers a very complete discussion around causation and causal inference, both from a statistical and philosophical perspective.
I'd like to add the following references (roughly taken from an online course in epidemiology) are also very interesting:


*

*Swaen, G and van Amelsvoort, L (2009). A weight of evidence approach to causal inference. Journal of Clinical Epidemiology, 62, 270-277.

*Botti, C, Comba, P, Forastiere, F, and Settimi, L (1996). Causal inference in environmental epidemiology. the role of implicit values. The Science of the Total Environment, 184, 97-101.

*Weed, DL (2002). Environmental epidemiology. Basics and proof of cause effect. Toxicology, 181-182, 399-403.

*Franco, EL, Correa, P, Santella, RM, Wu, X, Goodman, SN, and Petersen, GM (2004). Role and limitations of epidemiology in establishing a causal association. Seminars in Cancer Biology, 14, 413–426.


Finally, this review offers a larger perspective on causal modeling, Causal inference in statistics: An overview (J Pearl, SS 2009 (3)).
A: Almost surely in a well designed experiment.  (Designed, of course, to elicit such a connexion.)
A: Suppose we think the factor A is the cause of the phenomenon B. Then we try to vary it to see whether B changes. 
If B doesn't change and if we can assume that everything else unchanged, strong evidence that A is not the cause of B. 
If B does change, we can't conclude that A is the cause because the change of A might have caused a change in the actual causation C, which made B change. 
A: At the heart of your question is the question "when is a relationship causal?" It doesn't just need to be correlation implying (or not) causation.
A good book on this topic is called Mostly Harmless Econometrics by Johua Angrist and Jorn-Steffen Pischke. They start from the experimental ideal where we are able to randomise the "treatment" under study in some fashion and then they move onto alternative methods for generating this randomisation in order to draw causal influences. This begins with the study of so called natural experiments. 
One of the first examples of a natural experiment being used to identify causal relationships is Angrist's 1989 paper on "Lifetime Earnings and the Vietnam Era Draft Lottery." This paper attempts to estimate the effect of military service on lifetime earnings. A key problem with estimating any causal effect is that certain types of people may be more likely to enlist, which may bias any measurement of the relationship. Angrist uses the natural experiment created by the Vietnam draft lottery to effectively "randomly assign" the treatment "military service" to a group of men. 
So when do we have a causality? Under experimental conditions. When do we get close? Under natural experiments. There are also other techniques that get us close to "causality" i.e. they are much better than simply using statistical control. They include regression discontinuity, difference-in-differences, etc.
A: I noticed that 'proof' was used here when discussing the empirical paradigm.  There is no such thing.  First comes the hypothesis, where the idea is advanced; then comes testing, under "controlled conditions"[note a] and if "sufficient" lack of disproof is encountered, it advances to the stage of hypothesis...period.  There is no proof, unless one can 1) manage to be at every occurrence of said event [note b] and of course 2) establish causation.  1) is improbable in an infinite universe [note infinity by nature cannot be proven].  Note A; no experiment is taken under totally controlled conditions and the more controlled the conditions are the less the resemblance to the outside universe with apparently infinite lines of causation.  Note b; mind you, you have to have described said 'event' perfectly, which presumably means a perfectly correct language=presumably not a human language.  For a final note, all causation presumably goes back to the First Event.  Now go talk to everyone with a theory.  Yes, I have studied formally and informally.  At the end; no, proximity does not imply causation nor even anything other than temporary correlation.  The timespan of a mountain (given that they are alive; prove they aren't) and therefore the perception...is not that of a (wo)man.
A: There is also a problem with the opposite case, when lack of correlation is used as a proof for the lack of causation. This problem is nonlinearity; when looking at correlation people usually check Pearson, which is only a tip of an iceberg. 
A: Your example is that of a controlled experiment. The only other context that I know of where a correlation can imply causation is that of a natural experiment. 
Basically, a natural experiment takes advantage of an assignment of some respondents to a treatment that happens naturally in the real world. Since assignment of respondents to treatment and control groups is not controlled by the experimenter the extent to which correlation would imply causation is perhaps weaker to some extent.
See the wiki links for more information controlled / natural experiments.
A: In my opinion the APA Statistical Task force summarised it quite well

''Inferring causality from nonrandomized
  designs is a risky enterprise.
  Researchers using nonrandomized
  designs have an extra obligation to
  explain the logic behind covariates
  included in their designs and to alert
  the reader to plausible rival
  hypotheses that might explain their
  results. Even in randomized
  experiments, attributing causal
  effects to any one aspect of the
  treatment condition requires support
  from additional experimentation.''
  - APA Task Force

A: Sir Austin Bradford Hill's President's Address to the Royal Society of Medicine (The Environment and Disease: Association or Causation?) explains nine criteria which help to judge whether there is a causal relationship between two correlated or associated variables.
They are:


*

*Strength of the association

*Consistency: "has it been repeatedly
observed by different persons, in
different places, cirumstances and
times?" 

*Specificity  

*Temporality: "which is the cart and
which is the horse?" - the cause
must precede the effect 

*Biological gradient (dose-response
curve) - in what way does the
magnitude of the effect depended
upon the magnitude of the (suspected) causal variable? 

*Plausibility - is there a likely
explanation for causation?  

*Coherance - would causation
contradict other established facts?

*Experiment - does experimental
manipulation of the (suspected)
causal variable affect the
(suspected) dependent variable 

*Analogy - have we encountered
similar causal relationships in the
past?

A: In the twins example it is not just the correlation that suggests causality, but also the associated information or prior knowledge. 
Suppose I add one further piece of information. Assume that the diligent twin spent 6 hours studying for a stats exam, but due to an unfortunate error the exam was in history. Would we still conclude the study was the cause of the superior performance?
Determining causality is as much a philosophical question as a scientific one, hence the tendency to invoke philosophers such as David Hume and Karl Popper when causality is discussed. 
Not surprisingly medicine has made significant contributions to establishing causality through heuristics, such as Koch's postulates for establishing the causal relationship between microbes and disease. 
These have been extended to "molecular Koch's postulates" required to show that a gene in a pathogen encodes a product that contributes to the disease caused by the pathogen. 
Unfortunately I can't post a hyperlinks supposedly beCAUSE I'm a new user (not true) and don't have enough "reputation points". The real reason is anybody's guess.
A: Correlation alone never implies causation.  It's that simple.
But it's very rare to have only a correlation between two variables.  Often you also know something about what those variables are and a theory, or theories, suggesting why there might be a causal relationship between the variables.  If not, then we bother checking for a correlation?  (However people mining massive correlation matrices for significant results often have no casual theory - otherwise, why bother mining.  A counterargument to that is that often some exploration is needed to get ideas for casual theories. And so on and so on...)
A response to the common criticism "Yeah, but that's just a correlation: it doesn't imply causation":


*

*For a casual relationship, correlation is necessary.  A repeated failure to find a correlation would be bad news indeed.

*I didn't just give you a correlation.

*Then go on to explain possible causal mechanisms explaining the correlation...

A: I read all the answers. Some useful insight are given but no one answer seems me decisive, neither the accepted one (it can be correct but too vague).
I offered here (Under which assumptions a regression can be interpreted causally?) an answer about:

Under which assumptions a [linear] regression can be interpreted causally?

I think that it give a decisive answer to the question.
Now, it can be showed that any linear regression coefficients can be converted in a linear correlation (total or partial). As a consequence in my linked answer there is also the answer to the question:

Under what conditions does [linear] correlation imply causation?

Finally, if we are interested in in non linear relationships, the core of the answer remain the same but the math become harder.
A: If you want to determine whether $X$ causes $Y$, and you run the regression
$Y = bX + u$
Then $b$ is an unbiased estimator of the causal effect of $X$ on $Y$ (that is $\mathrm{E}(b)=B$) if and only if there is no correlation between $X$ and $u$, that is $\mathrm{E}(u|X)=0$. This is because $u$ can be thought of as anything else that causes $Y$. And so if this assumption holds, b is an unbiased estimate of the effect of $X$ on $Y$ ceteris paribus (other things being equal). 
Being unbiased is a desirable property of an estimator, but you would also want your estimator to be efficient (low variance) and consistent (tends in probability to true value). See Gauss-Markov assumptions. 
