The sunshine must have got into my head - I want to try & cast a more positive light on this.
Thinking about this from DH's point of view, he did something crappy a long time ago. Having idiotically missed his chance to come clean at the time, he then lumbered himself with having to keep a shameful secret from you throughout your wedding, DCs and all that's happened in between. When he made his mistake you were a couple. During the intervening time, you've become a family. His feelings have no doubt changed, as I'm sure yours have, and your lives have become much more closely intertwined. He has done all this with the secret lurking around in the back of his mind. It must have been difficult at certain moments.
He missed other opportunities to tell you. He should have taken them, but he didn't. Now he has - and you've found out that all the intertwining, the family-building and shared history contains a lie. It hasn't been built on a lie, but it contains one that you didn't know about before. Therefore you've been at an emotional disadvantage; of course you feel cheated and of course you question how much of what you've built up is real.
It's sort of like building a house together, if he knew there was a fault in the foundations and never told you. Your house will probably not fall down, but it's less solid than what you believed you were building. By hiding the fault, he robbed you of your right to decide how & where to build. If you don't feel angry about this, you should!
People who haven't had to live through such a let-down often talk about getting closure, getting over it and moving on. They think you should go (allegorically) "Oh, our house is on shaky foundations and I never knew? Well, never mind, it's still a lovely house, nothing's changed!" Things have changed - your house is still the same but now you know its hidden fault; it might fall down one day. From now on, you'll never feel quite so safe in that house.
But ... all other things being equal, you probably would have chosen the house even with the fault. Everything else about it is still good, so what's changed is your feeling about it. That's because your feeling was based on incomplete knowledge. So the first question is: how much of a deal-breaker is it? Not in anybody else's eyes, but in yours. Some people have a zero-tolerance approach to lies and/or infidelity. If you find you are one of them, then you're probably looking at divorce, either now or later, because you'll be incapable of loving someone who does this.
Very few people are zero-tolerance, though many think they are. If you can love someone who admits to having cheated in a past relationship, or you have lied about how much a pair of shoes cost, then you have it in you to be flexible. The most important thing for you to do is get pure clarity on what your feelings are and how they have changed. Don't expect that to happen instantly; there's a kind of grieving process to be suffered, for the honest marriage you thought you had.
For DH, there's a more complex task ahead. He's spent six years living with his secret. It's probably made him feel a bit crap, but it has also given him a kind of upper hand in your relationship. Can he see that? More importantly, can he give up wanting to control your knowledge about your own husband and your marriage? If he can, and wants to, he's going to have to work pretty hard to convice you he means it. He might have to tell you everything - every last, little thing - about where he goes, what he does and what he thinks, for as long as it takes. Does he believe he's able to give up control for your sake?
If there are any more skeletons in his cupboard (or mouldy sandwiches in his briefcase) he'd better fetch them out now, because a fresh start is what you're aiming for and you won't be able to do it twice. While you're at it, bring out any of your own! Do it properly :)
OK, this all sounds like hard work and it is. The reason I say it's positive is this: Very few marriages are genuinely founded on honesty and understanding. What you've got here is a chance to create the kind of marriage we all want; a marriage between two people who know each other as well as they know themselves, with 100% mutual respect and liking, as well as love. It's something worth going for imo.
I'm really sorry this has turned out so long! If things have changed while I've been typing, feel free to ignore - whatver happens, be gentle with yourself and good luck :)