Paint to cover rust in sink?

I have an old sink (about 50+ years old) in my upstairs bathroom. I'd like to see if there is a paint that would work to cover up the rust stain around the drain. Other than being old, the rest of the sink and faucets are in good condition and work well. Yes, I could just repace the whole thing, but what fun would that be?

Has anyone had success with painting a sink? I'd appreciate any helpful ideas, thank you so much!