Portrait of Jennie (1948) starring Joseph Cotten, Jennifer Jones, David Wayne, Ethel Barrymore, Cecil Kellaway, Lillian Gish
From IMDb: Eben Adams is a struggling artist in Depression-era New York City who has never been able to find inspiration for a painting. One day, after he finally finds someone to buy a painting from him, a young girl named Jennie Appleton appears and strikes up an unusual friendship. Each time they meet, Jennie is several years older, telling Eben she's hurrying to grow up so that they can be together.
Here's the movie trailer: http://www.youtube.com/watch?v=pYmtN8aG3zQ