I'm coming at the same situation, from the opposite side that you are. I'm clinically depressed and have been a total downer for my partner, so she's threatening to leave me - and rightly so. The thought of losing love is almost beyond my coping skills, and I'm always fighting with the idea that it'll mean the end of my life. However, I'm aware of my condition and realize that telling her what's going through my mind will do nothing to further my cause, and will actually just compound it by ruining the life of someone I actually love. Once you take it to that level, you'd better be prepared to face some consequences.
If you're in the middle of a break-up and you're told that they'll kill themselves, then you should probably tell them that you're going to inform the police of their intent to do harm to themselves. You really don't have a more logical option than that, because the only other options are to either let that person commit suicide and then resign yourself to life-long guilt, or to comply and resign yourself to a life of pretending to love a depressed person with control issues... both of which are totally self-destructive.
Seriously, take it from a person that's lived with chronic depression for decades; if you're seriously over with the relationship, the best thing you could do, for both of you, is to report it to someone who can help that person deal with their issues in depression, because that's the root of the problem - it's not you.