Hey, we all make mistakes. Cheating is bad, obviously, but at the very least you are taking responsibility for your actions and seem to have learned from your mistake. That’s something.
All you can do is tell your ex what you’re telling me — that you’re truly sorry and have learned some important lessons from your indiscretion. Be prepared to tell him exactly what those lessons are, and how you think they will prevent you from cheating again. He will need some assurance that you’ve really changed and won’t burn him a second time. Fool me once and all that.
You might also offer to go to couples counseling with him if he thinks it could help (and it can). You don’t say how long you two were together, but if it was a longer-term relationship, couples therapy can almost certainly help.
I’m sure you realize there is a very good chance he won’t take you back no matter what you do, which is his prerogative. You can’t really blame him if he does. Part of owning up to one’s actions is quietly accepting the negative consequences that go along with those actions. You blew it and now you might have to pay the price.
On the other hand, don’t spend the next year kicking yourself in the ass for what you’ve done, either. Everyone screws up. Learn what you need to learn from it, resolve to do better next time and move on. You have to forgive yourself, even if he doesn’t, or it will hinder all your future relationships.