What kind of sink, could you post a photo to help us better determine a suggestion?
A friend of mine used duct tape. It worked but did not look good at all.
Duct tape works on everything!
Hi. You can use an epoxy filler (think automobile repair) and an epoxy paint to go over the entire sink. It's a lot of work, but it can be done. In the meantime, I've never used it, but FlexTape advertises it sticks to anything and holds water. Good luck!