According to Grammar Girl, pronouns that come after prepositions in prepositional phrases are objects, not subjects, and as anyone who had my fifth grade English teacher knows, me is an object pronoun.
BUT, Slate's Lexicon Valley podcast points out that this isn't necessarily true, because of something having to do with Noam Chomsky and Middle English. It was a pretty boring story, but I came up with a cool tip for remembering which is correct.
Don't say between you and I. It's wrong. But don't say between you and me either. Instead use the grammatically unambiguous "tell anyone and I will cut you."