Yet another relationship advice question - long, but please help me out!

OK quick background – my boyfriend and I have been dating for a year (he’s 23, I’m 20) and our relationship has been fantastic overall, aside from the issues that have popped up recently.

Basically, he has been seeing his old girlfriend (of three years, they broke up in 2010) BEHIND my back! So far it’s only happened twice that I know about, but that’s enough to make me quite upset. The first time was in September – I actually asked a question about it: http://www.fluther.com/150296/how-should-i-handle-this/… anyway, after a big argument the issue was sort of resolved – he explained that she does in fact know about me and that they are JUST friends and I have nothing to worry about. So that was that.
This past week, I was getting weird vibes, so I’m not too proud to admit that I snooped through his phone to find text messages from/to her. Turns out he drove an hour to see her (in the town that I go to school in, I might add) and totally lied to me about it. Their text messages are so flirty it makes me sick. So I messaged the old girlfriend on Facebook. She responded assuring me there is nothing going on between them and that they like to meet for coffee every few weeks to “catch up”. Fine.
My problem is that I can’t stand him doing this behind my back. He says he only snuck around because he knows I’d be upset. OF COURSE I WOULD! And if he knows that, why do it in the first place?? Next problem – Originally I gave him an ultimatum – I know, probably a bad idea, but I told him he needs to pick her or me because it’s just not fair to me. Then I told him that he doesn’t have to cut her out completely, just talk to her every now and then and not see her. But he doesn’t seem to want to compromise at all. He wants to keep her and keep me; have his cake and eat it too?
I don’t know what to do anymore. I don’t want to lose him, but I can’t handle him being hung up on this girl. He says he’s not and any feelings for her are in the past—I’ve never met her so what am I supposed to believe? I hate him being so secretive and I just hate how he feels the need to spend so much time with an ex. PLEASE HELP.