He admits that what he did was wrong, but he will not say that they had a full affair. Which of course he has. I realise that some might say to believe him, but i can't believe him...who in their right mind would?
Perhaps he's one of the many men who can compartmentalize sex and love. Perhaps he had sex with her but because he didn't love her, he denies it was an affair. *shrugs* There are many men who don't think they're cheating if they are just using someone as a warm wet place to masturbate.
As for advice on how to get him to believe you're done with him: Heres what I think: Get a lawyer and start devorce proceedings. The first letter from you lawyer should drive it home that you're done. I would imagine he's used to you saying things but no following through and he's likely thinking this is just another one of those times. You need to have a proper visitation schedule and child support payments arranged anyway so a lawyer letter would be a good place to get that started.
“The willingness to accept responsibility for one’s own life is the source from which self-respect springs.” ~Joan Didion