If a piece of the clasp has come off, rather than a snapped pin, it's a tough one. I would say there is zero chance Rolex would be interested in doing anything apart from selling you a new one. (I tried them once with a clasp that was slightly 'sprained' and they weren't interested.)
Seth is your man, I would suggest, although I would suspect the broken part needs to be replaced. I can't see how it could be repaired.