None of us are perfect, and we've each done things we are regretful of. Ask for her forgiveness.
If your friend will hear you out in person, that is best. Take her by the hands, look at her in the eyes, and confess your humanity. Let her know that you failed her as a friend. If she won't meet with you personally, then write your message down in a card. Express sincere remorse and ownership for your behavior, regardless of being under the influence. After all, it was your choice to drink to excess.
Express what she means to you and ask for her ideas on how to make this right. Be willing to offer up some ideas. Make sure she knows it will never happen again -- and deliver on that promise! Trust isn't rebuilt in a day. Ultimately, convince her that you have learned something important (how much her friendship means) and you are trying to be a better person and a better friend.