Apologizing is part art and part psychology. Making your best case for "forgive and forget" means you need to get into the other person's mind and say everything she's already thinking. But before you make an attempt at an apology, make sure:
You're actually sorry for what happened
You want forgiveness
You're willing to work at forgiveness
You will graciously accept the other person's decision
Once you've taken those steps, here are six things you need to do to have the best shot at an accepted apology:
1. All in
Ask her if she really wants to know everything involved with the wrongdoing, and if so, why. But warn her there is no going back; once she knows she cannot un-know. At this point, if she agrees, admit everything, not just what she found out about. This step is the most difficult and can cause the most problems. She will hear things she likely doesn't want to hear and will be hurt all over again. But once the truth is fully out, she can begin the healing process and get past the hurt and anger.
Acknowledge that giving your apology does not absolve you of wrongdoing. You understand it doesn't right the wrong, and neither does the apology's acceptance. You are apologizing because it's the right thing to do, and she deserves it.
3. Help me help you
Have her help you put a plan in action. You can come up with a long-term plan to make amends on your own but would appreciate her input. Ask her to include things that are particularly important or special to her, so she knows you care about her feelings and needs from this point on.
4. Always, forever
Let her know you are willing to do anything (legal and without harm) to earn back her trust, with no time limits. However long it takes, you are willing to persevere. You did the crime, so now you want to do the time.
Admit to her the betrayal was completely your choice and had nothing to do with her. Even if the motivation to do wrong was brought on by something she did or didn't do or say, keep the focus on your actions and feelings, and your ultimate decision to act inappropriately in response to those feelings. You could have talked through it; you could have gotten outside help. But you chose a deceitful and hurtful path, and that's no one's fault but yours.
Outline what you should have or could have done instead of what you did. Make sure she knows you didn't do your best, and you weren't thinking of her in your weakest moments. Whatever pain you feel from this situation is deserved, but hers is not. Let her know she deserved better. And even though you don't deserve her now, you want to do anything and everything to earn that deserving again.
If you are giving the perfect genuine apology, you want to make it impossible or even painful for the wounded party not to forgive you. You want her to see only benefits in forgiving you and moving forward, and give them no reason or motivation at all to dwell on the past. But be warned, you must mean everything you say. Apologies are for the sorry and ready to change, not for the manipulative and intent on remaining the same.