I know what *I* would do in that situation, but... that is probably not what *you* ought to do, if only because you likely have different circumstances.
To put it simply, cheating on a relationship is the worst sin in the world for me short of "the big three" (Murder, Rape, Torture). For that reason, my normal instincts would push me to the point of extreme anger at first, though I probably would be good enough to conceal this from the person in question. It would come out later when I was somewhere all alone, just as all my other "extreme" emotions do (Joy, Sadness, or Anger).
My first instinct from there would likely be revenge or some other very sinister action that would basically be a "break-up and more" idea. Considering my past history, the fact that anyone would have cheated on me likely would prompt such a response, because I have a history of making examples out of people to hold up to others in order to say "don't do what this idiot did." Chance are however, i'd only take this approach if the person who cheated appeared to show a lack of shame for their actions. They showed no care for your feelings, and so I in turn would show no care for their livelihood. That is a vicious road to cross.
At the same time, such thoughts are unlikely to yield a really good result, and this is something I know well. Revenge is an ugly business. Also, since this is a case of someone seeming to feel bad about this, vicious revenge isnt exactly the best option. So, once i had cooled off a bit and had a chance to relax once things had sunk in, i'd probably assess what occured.
There are basically two real options. You can tell them to go away. Simple, done deal. How much you would let it be known as to why you decided to break it off is at your discretion, as well as based off of an understanding of how the other person is. If they are a spiteful type, then i'd be a little wary of going too far, but thats a given.
But the other option... and its the one that I think fits this case well, is to keep things going, if only for a while. If this person really did come forward so quickly, then they clearly do feel bad about it. If things are to continue, chances are the other person will have to be on some kind of "probation" of sorts, and they will have to know it to be as such. Nobody walks away from such action lightly, but it is up to you and your specific situation to determine just how to carry that out. In short, keep the person, but don't let them forget what they did.
In the long term though... it depends I guess on your own comfort zone. I've never been cheated on, or really had a serious relationship, but I do know that I am someone, thanks to past events, that trust is extremely difficult to acquire with. Wven with people I do trust, I am always asking myself whether or not I should. I am someone who is prepared, possibly resigned, to a feeling that someone will abandon or take advantage of me sometime in the future. When people actually do such things, I have in some cases managed to maintain a friendship with them, but the trust I placed in them is permanently shattered, and a loss of trust from me basically is a loss of true honesty. And without true honesty from me, a person is inevitably confined into a place where they know little to nothing of what I really think or believe. Its a death sentence for that relationship, and its a place that people don't come back from in my world.
Because of that, chances are if someone was to cheat on me, that relationship would probably linger on for a little while, but it would probably never develop into anything serious or go into the long term. I find people who cheat on others to be far too morally disgusting for me to want to be intimate with such people, no matter how guilty that person feels.
That's just my opinion.
- Rei