What is the reason for this discrepancy between confidence level and probability? I ran the following R code: 
library(Hmisc)
binconf(x=1, n=70)
#   PointEst         Lower      Upper
# 0.01428571  0.0007327613 0.07658187

Which is the confidence interval with 95% confidence.
But when I try to find the probability:
pbinom(0.07658187*70, 70, 1/70)-pbinom(0.0007327613*70, 70, 1/70))
# [1] 0.634254

It gives 0.63; that is, 63%.  Why is this so far from 95%?
 A: It isn't clear that the output is incorrect.  The problem is that there is a confusion about the distribution and the interval you are referring to.  Specifically, the binconf() function pertains to the sampling distribution of the proportion.  The code you used to double check that is based on a misunderstanding.  
In general, there are three distributions that we are working with or thinking about when we analyze data.  These are the population distribution, which is ultimately what we care about; the sample distribution, which is what we really have access to; and the sampling distribution, which is what allows us to connect the former two.  
Since the type of data you have come from Bernoulli trials, your data are distributed as a binomial, and the parameter that governs the behavior of a binomial is the probability of 'success'.  From your sample data, you calculated a mean / the proportion of successes (i.e., 1/70 = 0.01428571).  That can be used as an estimate of the binomial probability.  Just having a point estimate isn't enough for most people.  It's nice to have a sense of the precision of the estimate.  The confidence interval can give you a sense of how close your estimate might be to the true probability.  It is not the middle 95% of the population distribution, though.  Instead it is the middle 95% of the estimated sampling distribution.  
To calculate an interval covering the middle 95% of the population (if you wanted that for some reason), it would be:  
qbinom(c(.025, .975), size=70, prob=0.01428571)
# [1] 0 3

That is, approximately 95% of values drawn from ${\rm binom}(0.014, 70)$ will fall within the interval $[0, 3]$.  
The function you used (e.g., pbinom(0.07658187*70, 70, 1/70)), isn't really appropriate here.  It has to do with the proportion of the ${\rm binom}(0.014, 70)$ distribution you have passed through when you reach the quantile 5.360731.  This isn't very clearly related to anything you are trying to do.  
