At this point, perhaps we can consolidate the comments above into an answer. If you agree with the point made by @cardinal and @Procrastinator that the probability of finding a value at your endpoint is 0, and you are not concerned about this (it doesn't matter to you that you won't get this value), we can set @whuber's code aside, and just recommend runif().

Since I'm mostly repackaging ideas from several people here, I am marking this 'community wiki' (i.e., I don't need to take credit for it).

Let me know if you have any other concerns.