Fiancé doesn't trust me after I cheated to get back at him

I slept with another guy to get back at my fiancé after a girl claimed to have had sex with him.
It all turned out to be lies and I’ve regretted it ever since.
I’m 22 and my fiancé’s 23. We’ve been together for three years and have a beautiful three month-old son. We’d only been together a couple of months when I heard rumours that a girl was claiming to be having sex with my fiancé.

I was furious and slept with another guy as payback. My chance came when I was invited to go on a friend’s hen night. We had a meal with lots of wine and then went to a club where I met this other guy.
He had looks to die for and I flirted with him like mad. We sneaked off together and had sex round the corner from the club down a dark alleyway.
Next morning I wasn’t proud of what I did, but at the time I didn’t care.
I thought my fiancé got what he deserved but then it turned out that the girl who had claimed to have had sex with him was lying. She fancied him herself and was spreading rumours to try to split us up.
I confessed to my fiancé because I felt so guilty. I want him to be able to put the past behind us and move on, but he doesn’t trust me and keeps accusing me of things I haven’t done.
He accuses me of cheating even while he is at work. Now we’re arguing all the time. I don’t want that in front of our baby. I want to be a family and for him to be happy.