What is the name of the movie about a woman who steals peoples faces?

I'm trying to locate the title of an old black and white either sci-fi or horror film. Its about a woman, possibly an alien, who takes people's faces. At one point she takes the face of a nurse and attempts to attack a young man in his room by using her fingernails to hypnotize him. By the end she is either pushed off of or falls off a cliff, and the credits are shown rolling over her dead body. If anyone can give me any information about this, I would greatly appreciate it.