Hi, I'd like to help you with your question.
It is great that you want to take responsibility and deal with this issue. Trust is the foundation of a marriage and you need to let your husband know so you can confront this problem and rebuild the trust in your marriage.
Have you consider seeing a therapist with your husband about this issue? If you tell him in counseling, you have the counselor to help you both maintain your composure and work through your feelings about this. You can ask your doctor for a referral to a therapist or talk with your pastor if you attend church. You can also search on line at http://therapists.psychologytoday.com/rms/
Seeing a therapist will also give you the opportunity to work on solving the root problem of why you over spend your budget and why you hide it from your husband.
If you chose to confront your husband without a counselor, you may want to approach this gently. Do not just blurt out what happened but lead up to it slowly. Start by saying that you have something you need to take responsibility for. Ask him for his patience. Tell him you will be fixing this problem yourself, but you need to let him know about it. Then tell him the problem. You may have to deal with his reaction being the same as before. What you need to do is let him vent. Once you can talk with him again, explain how you are going to fix this problem. Part of his reaction might be the responsibility he feels is suddenly placed on his shoulders as a result of the debt. It is no longer just your problem, now it is his too. And he may feel he can't trust you. Because of these reasons, you need to have a plan to take responsibility. Get another part time job to pay for the debt. Set up payments so you can afford to repay what was lost. Whatever it takes for you to repair the situation. Then follow through.
I hope this helps you,Kate