There are quite a lot of medical cases where people have damaged various parts of their bodies. Where other body parts (not the brain) have been damaged, an amputation, a transplant, or a cure, does not alter the awareness of the patient. Sure, he likely won't live like he did, but he still would think, behave, retain memories, like he did. His life would continue that way.
However, where brain death is concerned, I honestly haven't read about a case where the patient regains consciousness, remembers his past, and knows who he is (was). That's not sufficient scientific evidence, but probably sufficient circumstancial evidence to claim that awareness lies in the brain?
The brain, like any other organ, ages with time. I'd hazard a guess that even if brain transplants can keep the owner of the brain alive through various younger bodies, it won't make him immortal. The brain will eventually wear out and die.
Thanks for the valuable comment. I am very curious about the outcome is such a procedure could be successfully achieved.