Which is proper grammer? "A" Hour or "An" Hour?

A friend and I got into an argument about this the other day. The general rule is that you use "an" with vowels, but "H" obviously isn't a vowel. I said that its "An" Hour because the "h" is silent and it sounds like it starts with a vowel. Its a exception to the rule. My friend said that it "A" hour, of course, citing the rule. Isn't it a shame that I, a law student, and he, a medical student, seemingly smart people, couldn't come to a conclusion on this elementary issue.