If we disbelieve $H_0$, why quote a p value calculated assuming $H_0$ was true? Hypothesis testing seeks to reject a null hypothesis ($H_0$) on the basis of an assumption made about the sample following a certain distribution. This assumption is conditional on $H_0$ being true. So what is the flaw with the following reasoning?

If I reject $H_0$ in favor of the alternative hypothesis ($H_1$), I
  must then assume that (1) $H_1$ is true, and (2) that at the
  same time the variable under examination follows the distribution
  used to reject $H_0$. Since we also stated that the assumed
  distribution is conditional on $H_0$ being true, it appears that $H_0$
  has got to be true in order to justify the assumption made about the
  distribution. Logically, it would therefore seem that both $H_0$ and
  $H_1$ have to be true, which is a paradox, because they are mutually
  exclusive!

 A: (Started as a comment, but it's much too long)
Let's consider this a different way. A more general version of the question is --- can we use reasoning involving conditional probabilities when the thing we condition on is false?
It's not simply permissible -- it's necessary.
Consider this in the context of Bayes theorem:
$$P(A_i|B) = \frac{P(B|A_i)\,P(A_i)}{\sum\limits_j P(B|A_j)\,P(A_j)}$$
Note that the $A_j$ are mutually exclusive (and exhaustive). All but one of the  conditionals in the denominator must pertain to a condition that doesn't hold - but that doesn't imply that reasoning involving those conditional probabilities will be invalid -- Bayes' theorem is true as a result of us reasoning using conditionals that condition on events that we know don't hold.
The conditional probability $P(B|A_j)$ is a perfectly valid conditional probability, whether or not $A_j$ actually obtains.
It's perfectly okay to reason via conditional probabilities that relate to 
conditions that don't hold; the results are logically valid. [Indeed, I bet you do it constantly without any concern.] 
For example, if I say "Alison would have her umbrella if it were raining" and use this plus some data to support a conclusion: "She doesn't have her umbrella, so it's not raining", my conclusion doesn't become invalid because the conditional was untrue (The fact that "it's not raining" doesn't endanger the truth of the conditional that reasoning was based on: "if it were raining"). 
A: The principle is like a "fuzzy" version of the contraposition principle (or reductio ad absurdum principle, I'm not sure). 
Consider that every dog has four legs. Then if you sample an animal with two legs you are sure it is not a dog.
Now only consider that every dog has a high probability to have four legs (in other words a high majority of dogs have four legs). Then if you sample an animal with two legs you conclude it is unlikely a dog. This is the principle of hypothesis testing (but in practice it requires a sensible choice of the event having high probability under $H_0$).
A: You have a misunderstanding.  The 'alternative' hypothesis ($H_1$) is simply the negation of the null hypothesis.  When conducting, say, a power analysis, we will specify a specific sampling distribution around a point estimate (for example a mean treatment effect) that we believe in, but rejecting the null does not make that point estimate true.  Based on the logic of hypothesis testing, the alternative hypothesis is not that point estimate, it is just the negation of the null.  There is no particular sampling distribution of a test statistic that is associated with the negation of the null.  
In addition, the meaning of the $p$-value is predicated on what may well be a counterfactual premise.  The $p$-value is the probability of getting a test statistic as far away from your null point value for your parameter (or further) if that point value were true, whether it is actually true of not.  Even if the null isn't true, it can be true that the test statistic would have the specified distribution under the null.  
You are striking on an important insight, though.  Once you no longer believe the null obtains, it is no longer clear what meaning the $p$-value has to offer.  
