How do you break up with someone suicidal?

This is linked to my other question, but from advice from family, friends and strangers alike I have decided that I would be happier being single and not being with him anymore because I can't forgive what he has done (telling another woman he loves her but telling me that he would cut off contact but never doing so). However, during the entire month of stringing me along, whenever I got upset and mentioned that I wanted to leave or breaking up would be better than feeling this way, he would get extremely upset and say he couldn't live without me and that he needed me.
When we agreed to go on a break, I'm the one that pushed that I should leave his apartment as soon as possible and I took all my things. He saw this as a break up and when I went back to pick up some things I had left I found that he had been crying all day and had self-harmed and attempted to slash his neck and had checked the room for places to hang himself. He really doesn't think he can live without me. Only one friend of his knows the full extent of this and that is because I told them so afterwards. He only calmed down when I reassured that it was just a break and that we hadn't decided to break up yet. But he can't keep doing this when we agreed that the break was to decide whether we should break up or not. It feels like he isn't actually giving me a choice but hoping that time apart will make me forgive him or miss him enough to go back. How should I approach this situation when I know that I would be unhappy being with him?
How do you break up with someone suicidal?
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