Definitely not the 60's but the 70's.
Spoiler ahead:
I do recall that the title sequence had a chase motif to it: as if to denote that a man was chasing a woman and vice versa....
At one point in the movie, the murderer (murderess?) is taken by the detective in a fast motorcar ride across town. He tries to prove that he is a great driver as the person he claims to be happened to be one.
The final scene has the detective in a room begging the woman to accept his claim. And finally she says "you can't be him...because I killed him". At this point, the detective goes to the phone, facing camera, with the woman in the background, and calls the police. He identifies himself on the phone as a detective...
Fade to black and ... the director comes out and asks the audience not to reveal the ending.
Anyone out there know this? C