How does rebirth fit with the fact that everything is impermanent?

One of the lessons Buddhism teaches is that everything is impermanent. Our thoughts, our feelings, our emotions, our bodies, our life. Everything. We are surrounded by death. We are mortals. There is an end to this. However, Buddhism also has the concept of rebirth. Karma follows you not just in this life but in your next. It's also said that the Buddha gained the ability to recall his previous lives. This to me implies some sort of a soul (for lack of a better word), condition, or consciousness that is not impermanent. After you die you get reborn. There is an "you" that transcends the current (impermanent) existence. I guess, my question is, is there some sort of a soul, consciousness, or a condition, that is not impermanent? When one reaches enlightenment (nirvana) and stops the cycle of rebirths, what happens to them?