Well the version I watched.. The jokers is seen falling and laughing manically as he falls. Of course we don't see him hit the ground
.. But one assumes from that height, it is implied that he dies.
Your version must have cut off. Batman catches him as he is falling and pulls him back up. They talk about how they'll probably be doing this forever, and Joker sort of derides Batman for refusing to kill out of some "misplaced sense of heroism" or something. Joker mentions what he did to Harvey Dent, prompting Batman to run off as the police arrive to apprehend Joker. Next and final scene, Two-Face with Gordon's family.