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what's the difference between a friend of me and a friend of mine?

They would mean the same. However a friend of "me" is not correct. You should use a friend of "mine".

"A friend of mine" is used, as in, "She is a friend of mine."

"A friend of me" isn't really used.

I would suggest "my friend" or "a friend of mine."

This is how I understand it:

1. In 99% of the cases, only "He is not a friend of MINE" would be correct.

2. But there are (rare) occasions when "He is not a friend of me" would be correct:

Let's say that you are the president of a club. Mr. X. does not like your leadership. Thus, you might tell a friend that "Mr. X is not a friend of ME." That is, he does not like your policies. It has nothing to do with personal friendship.

A friend of me is incorrect.
A friend of mine is correct.

Craig is right and his answer is the best one to remember.
A friend of mine is correct.
A friend of me is NOT correct and it is not worth your time remembering or using it.