I agree with 50guy's advice, but there's one important thing to add: why do you think you've been put into a position where, as you put it, you "don't know what you have to do to prove to him that you still want to be in this marriage."?

Sure, he feels guilty, feels like a failure, feels sorry. And yet through these actions he is continuing to knowingly cause you emotional distress.

When you boil it down, he, like any adulterer, chose to have an affair because, ultimately, he felt like it and at the time he chose to disregard the consequences. And now he's again doing what he feels like doing, acting with emotions and not rational thought, even though he knows it's making things worse for you.

There are two reasons for this that I can think of:
<ul type="square">[*] His emotions are overwhelming him and he can't help himself. If you believe this is true, then follow 50guy's advice and lay it on the line, telling him how this is affecting you and that you want it to stop so that the two of you can move forward.[*] The other reason could be that he resents having to end the affair and resents the situation he's in, and he's being passive-agressive and manipulative in ways designed to punish you for it. I hope this isn't the case, and only you can answer this question... but don't immediately discount it. Could this be possible with your husband? If it is... you may have a different kind of ultimatum to give him.[/list]