He'll do everything to work on the relationship?
What fucking relationship?!
He screwed it up and threw it in the bin quite a while ago really, didn't he?
I advise you from the bottom of my heart to get the fuck out of any so-called 'relationship' with a person like this. Two and a half years of lying to your face. Making everything a mockery. And, and - during the time you had your child together. There's really nothing to say about that except how terribly terribly sad that is - that wonderful, special time when you become a family, you have a brand new baby together that only you and he are the parents of, nobody knows and enjoys the little baby world you have together except you two - and that, that meant NOTHING to him. He could go straight from that to fucking another person. That's not redeemable. That shows like nothing else how defective he is inside, and I do not see how that can be changed.
If you stay with him I don't think you'll ever relax again - why the fuck would you? It's not some 'mistake'. It is what he is. Even if he is now truly sorry, realises how terrible his actions have been etc., is horrified - it doesn't change the fact that he IS that man. And who, who in their right mind would want to share a life with that man?
What would ever be natural, happy, relaxed, trusting about 'rebuilding' with him?
The fact that he's even begging tells you that it's not really sunk in, what he did. If it had, and if he had a shred of humanity and morality he'd be saying to you - I'm sorry and I know that no sane healthy person would choose to forgive and stay with someone who had done this to them, so I know we're over.
But he's telling you he'll make it up to you. By doing what exactly? The issue is that he is a man who is capable of the most horrific deceit, and who is capable of shitting on his nearest and dearest without a second thought. For years. No moment of madness. Now, he is still that man, yes? He hasn't had a personality transplant. So what exactly does he plan to do? What exactly does he plan to do about the fact that every time you look at him you'll be thinking 'You fucked us over in a way I'd not even do to my worst enemy.'?
I think you now how you will feel if you stay with him. Others have said similar and have talked of wasted years. Don't waste those years and don't waste your DC's childhood on 'building' a family with a hole at its heart.