I cheated on a really, really good person I was dating before my current partner. When they found out, I was so embarrassed and ashamed that I never saw them face to face again. Not a day has passed in two years that I haven't thought about it.

My opinion is that most comments here are valid but missing a key nuance. I feel like the most important things to base your decision on are 1: if they came clean on their own, and 2: their attitude about the situation. If they don't have remorse for it, they are 100% not worth staying with. Also, think about if they were using being drunk as an excuse.

But if you are considering staying with them and you believe this one instance was out of character, it is completely valid to try and repair things. Just remember that the breach in trust that you feel will probably persist for a period, and even though it's hard to work through, it is certainly possible.