Forty year old guy here, married just under six years.
Nineteen years is a long time; I imagine you've both been through some hardship in that period. And it certainly may be worth continuing to work on.
But I wonder: Is this the first time he's cheated in 19 years, or the first time you've caught him in 19 years? Maybe while you're being honest with each other, you can ask him that question.
Would I ever cheat? When I was 19, I would have said, "NEVER! I am an honorable man!" But doesn't everyone think they're basically a good person? I do. So I think people start convincing themselves some things are ok. "It's just a kiss. That's not cheating. And Zarelda is so lonely. Really, I'm just being helpful. I won't tell the wife because she might not understand." And it gets deeper and deeper.
So again, would I ever cheat? Maybe. I haven't. Yet. But I'm on guard against it. Because it has happened to too many good people, where they screw up and do what they KNOW is wrong, but somehow have convicned themselves it's ok. I didn't get married at 19. I got married at 34. I'd learned a few things about other people, and about myself. I know I can be tempted (usually by something containing chocolate...) and that while usually strong, there are times I'm weak. So I can't tell myself "I'll never do that." but instead, "That could be me if I'm not careful."
Forget what we (at Yedda) say: Is the pain you went through, and might someday have to go through again, worth the upside? Would it be better to have this guy gone -- or would it be better to have him, warts and all, in your life?
Finally, and I hate this part of my response, because it is so cliche, but it is true: Sometimes, it takes a third party to help you sort through all these questions: A counselor, either for you both or JUST for you, privately, can be a big help.
There's one last thing: You're at least as old as I am, so you know a bit about life. There's a good chance you've already decided what to do. It may be burried in your mind, but part of you has probably made the decision: I'm going to give this jerk another chance --- or, He's out of here. A neutral third party can help you hear what you may already said inside yourself.