OH dear.
Sadly I know a great deal about this kind of problem, as your h and mine could have been the same person until last year.
I never did 'know', as any evidence I found was denied and explained away with all kinds of bollocks which I cant now believe I fell for. But in fact he lied and did this behind my back for 5 and a half years (to my certain knowledge now). He was early 50 when he stopped, and the women were 18-30, with the emphasis on under 25 (I would say). My daughter is 25, and another 16.
I personally found the sex addiction (Patrick Carnes stuff) very illuminating, but in fact it doesnt matter whether it exists or not. And anyway, Carnes changed his views later on to talk about compulsion as opposed to addiction.
This means he uses sexual experience in order to feel better about himself, or to gain/exert power or control, or because he confuses sex and love.
However, he also knows that it is unacceptable, and that he is out of control to do this. He doesnt feel good about himself.
So why does he not stop?
For the same reason that a man who has a one off affair doesnt stop until caught, and only then if his world falls about his ears.
Because he cares about meeting his 'needs' more than about the marriage, and because he doesnt want to do the hard work of sorting out his issues. Rather, he prefers to escape from them. Sure, he would prefer you didnt find out and will half kill himself to hide it, but at the end of the day he thinks his needs are more important than yours.
I am afraid you have only one option, and it is easy for me to say this, because I havnt had to take it (yet).
If you are keen to keep your marriage (you may well not be) this is what I suggest.
Tell him that he needs to leave the family home and get counselling, and that you will not readmit him until you are convinced yourself that he has stopped, and for good. And that you want to attend a couple of his counselling sessions, too (though not all). That you want him to see someone with experience of his issue.
I didnt have to do this as my h had never previously been outed for this (through his lies), and when I did find out, through insisting on the truth I just knew I wasnt getting, it was after a few weeks at Relate and also after I had removed myself for some weeks from the bedroom and any involvement in his life.
He did stop, and sort himself out, but it was f'ing slow work, and I think I should have kicked him out for a while at the beginning to speed things up. There is absolutely no question in my mind, not a fragment, that if I discover this again, he is OUT, and I will tell all of his family and friends why, too.
In your case, you have caught him in this before, and quite clearly he simply doesnt believe that you have any boundaries to resist him. And currently, you havnt, have you?
Can I recommend Patrick Carnes books and also 'Boundaries: When to Say Yes, When to Say No, To Take Control of Your Life, by Cloud and Townsend.
Personally I read myself sick on the Carnes/Sex addiction/compulsion stuff, until a very clever mumsnetter pointed out quite rightly to me that I might help myself and him more effectively by sorting out how the hell I had got here in the first place (enabling, codependence and boundaries).
ie what was it about me and my life and made me put up with him for all of those years when I was suspicious but couldnt 'prove it', and when he was actually treating me more generally like a second class citizen.
This is the key issue for you, but you may be some way from this at the moment
Feel free to pm me.