Now that I've thought about it more, I think that you both got a point. This saying is taken out straight from the Bible, and the relevant passage in it does suggest the use you referred to. But it could be that people no longer remember the origins of this phrase, and therefore use it inappropriately, like me.
There's nothing objectionable with correcting people, C_B :) As much as I don't like to be wrong, I like being wrong twice about the same thing even less.
Edited. We seem to think along the same lines, Dontcry :)
I've always used it in the sense that the "spirit" is the soul, the holy spirit inside us or like the "good" angle on your right shoulder. The "flesh" is your human side, satan or the "devil" on your left shoulder.
The "spirit" is willing to be "good" and stay pure, but the baser, human "flesh" is weak and likely to give in to temptation.
I think other interpretations have come from a misunderstanding of the original saying.
What c_b said. I've only heard this used in the context of not giving in to temptation. I don't believe it is used in the way that Mia suggests (even though it would seem reasonable enough to do so). But "the flesh is weak" is a reference to the weakness of human nature in resisting temptation, I think, not actual lack of strength to perform some physical task.