Here's why:
* "She can sing like Marilyn Monroe": The most famous rendition of this song is indeed by Marilyn Monroe in the film.
* "Remember rest but a girl": The lyrics contain the line "They're like a girl's best friend, I'm told/They're yours until they're sold/But they're a girl's best friend, I'm told/They're yours until they're sold." This line, although not verbatim, seems to match the "rest but a girl" part you mentioned.
* "Popular song": "Diamonds Are a Girl's Best Friend" is a very well-known and popular song, especially thanks to Marilyn Monroe's performance.
While there are many other songs about diamonds, the combination of these factors strongly suggests "Diamonds Are a Girl's Best Friend" is the song you're looking for.