The debate usually starts like this:
Theist: I believe that God exists.
Atheist: Prove it.
Theist: You prove he doesn't.

Notice how the burden of proof has shifted to the atheist.

Actually, the atheist made the first mistake when he asked for proof that God exists. What the atheist should have done is ask the theist to describe God (otherwise neither party will know what they are talking about).

This is is not as easy as it sounds and if the atheist pushes hard enough, the theist will eventually reach the point where he declares that god is "infinite" and beyond the understanding of mere humans (incomprehensible, or ineffable).

When that happens, the atheist can say, "Touche. We don't know and cannot know anything about your god, not even if he exists, thus atheism is the most sensible position to take on this subject."

As you might expect, the theists have come up with many different ways to escape this dilemma (the Negative Way, for example). The escape routes can be easily refuted, but if you are not familiar with them, you might run into trouble, so be careful.