How can I regain my girlfriends trust?

Okay, here is the deal. We went out for quite some time. A year and a half almost to be exact. And I was happy with her. It was amazing spending time with her and everything, I even wrote her a song! One night though, we were talking on the phone and we were BOTH wondering, Is this just teenage romance? WIll this REALLY be forever? So we got into a little argument and I was honest and said I didn't know... I guess we got off the phone with both of us wondering. It just so happened, That my EX decided to come to my bedroom window to talk to me. I really didn't like her because she put me through so much pain. But she was crying and I saw it looked like she needed help. It was about how her dad was a drunk and he hit her and everything. So we talked about it and after she tried to kiss me. I asked what she was doing and she said she still loved me and everything but i told her I had a girlfriend already and I was happy. She wouldn't give up and had me on my bed and I gave in. We had and everything and I will admit I did want to. I know it was wrong. So the next day I saw my Girlfriend at school and I admitted it to her because I felt horrible for what I had done. I left her crying and for about a month, her eyes were bloodshot because she was crying so much everyday. I started to talk to her and I told her that I was honestly and truthfully sorry and that I DO love her. We talk a lot now. We aren't going out but I want to. We have held hands before and kissed when I took her to the movies on the last day of school. But when I asked her out she says she still feels betrayed and knows she will always have that sense of betrayal. What I want to know, is HOW can we get rid of that sense of betrayal? And HOW can I regain her trust?