Ok, this may sound extreme, but this is what I’d do if it were me:
If I were in your shoes, the first thing I would do is break it off with him. For the long term. Then I would tell him that if he really does want to change that he can get counselling, and be a good single father to our son. For me to consider being a couple, I would expect him to prove to me over the course of several years that he is trustworthy. By that I mean several years of him not seeing anyone else, and of he and I not dating either. We’d be in each others lives as parents and, so I’d have the chance to see a change.
On the surface, this idea of no dating anyone for several years sounds unreasonable, but I have heard many experts advise single parents not to date while their children are young. So given that you and he are both responsible for this little person’s soul, I think it would actually be a reasonable request, as well as a reasonable thing to consider for myself. In fact, while I would give him the option of showing me a change over many years, I would not “count on it.” Rather, I would get on with my life as a single mother, and plan for that to be my life for the next 18 years. If he changed, Alleluia. If not, I didn’t stay single for him, but for my child.
Of course, I also realize that just because I think it is a reasonable request, doesn’t mean that you or he would be willing to give up the search for another mate. In that case, I’d have to say it’s time to end the romantic part of your relationship now, as he is not trustworthy, and won’t be proven trustworthy for a very very long time if ever.