don't know the name of this movie!!!

I saw a movie a while ago which was abt a man and a woman who keep having chance run-ins. The last chance run-in he offers to walk the lady home. She then lies and points to an apt bldg and says she lives there and his reply is like: "you live in this bldg?" and of course she continues the lie by even going to an apt and sayin she lived in it. He plays along knowing that she is lying bcuz in actuality its his apt bldg and his exact apt she is claiming to live in. She even goes so far as to move in her clothes and pretend to live there as he comes by for "visits". The movie is in black and white and when I asked my co-worker he seemed to think it was a Loretta Young movie. I have searched every engine and cannot find it or anything close to it. Sum1 plz help me.