Here's the DVD cover for the 1959 film The Diary of Anne Frank, starring actress Millie Perkins as Anne. The film was nominated for best picture of the year (losing to 'Ben-Hur'). Shelley Winters (as Mrs. Van Daan in The Diary of Anne Frank) did win an Oscar as best supporting actress.
If Millie Perkins looks slightly old for the role here, it's not by chance. Perkins was born in 1938, making her 21 years old when this film came out; meanwhile, Anne Frank herself began writing her diary at age 13 and was 15 when she died in 1945. However, Wikipedia notes that Perkins "received almost universally excellent reviews for her portrayal of Anne."