The answer may be that you cannot trust her. Some people are terrible at keeping secrets and blab things, no matter how much they fight it. (Frankly, it's rare to find someone who will keep a secret entirely to themselves forever, but some are especially bad.) These people tend to describe themselves as "having no filter". I'm assuming your friend falls under this category since she immediately blabbed such a large secret. They can be great friends in every other way, so you don't necessarily need to cut her out of your life, but you may need to refrain from sharing any more of your deepest and darkest secrets.
What you can do is forgive your friend. Talk to her and let her know how you feel. Ask her why she did what she did, and give her a chance to apologize and try to make amends. If you need time, that's okay, tell her that. Once you are ready to forgive, do so. But forgiving doesn't mean that you will forget what happened or that you will trust her with secrets again; it means you are letting go of your anger and resentment, and that she doesn't "owe" you anything.
At this point, you can talk about sharing secrets again. This will depend a lot on her, what she's willing to do, and how honest (with both you and herself) she's willing to be. Is she generally terrible at keeping secrets, or was this a rare moment of weakness? Does she want to be responsible for keeping your secrets, or would she rather you shared that burden with someone else? Is she willing to improve? Is she willing to be your friend if you don't trust her with that responsibility?
If you both decide that you want to be able to share secrets with her, then I'd let her practice with smaller ones. If you decide it's best to keep some things to yourself, then share your problems in a more vague way in the future, so that she can't betray you but can still support you.
It's likely she will be offended if you tell her you can't trust her anymore, or if you refuse to give details on some things. Blabbermouths tend to blab their own "secrets" too, and they often don't understand why it's such a big deal when they do it to other people who keep things more private. It doesn't mean they're trying to hurt you or don't care for you; they just don't get the impact their actions have. I've had some success with getting them to understand how it makes me feel to have my secrets revealed, and sometimes they learn to keep things to themselves. But sometimes they can't or won't change, and you have to decide if you're okay with that or not.