Let's start with the bad news, rebuilding trust is very hard because whatever happened, happened, and you can never take back time. What's done is done. Period.
When trust is broken something real is lost, that's why when people lose a loved one they feel like a part of their soul was ripped out. Because our relationships are real. And trust is what makes them real.
What's even harder for people to accept is that forgiveness is more like a gradient rather than a boolean. Forgiving is more of an intention than an actuality; sometimes it works sometimes it doesn't. Sometimes the offended person decides to forgive just to realize later that some things can not be forgotten, there are always things that remain in the background -and even if everything that remains is just a memory, recollecting that memory will always come with a bit of bad feeling or emotion attached. Although we don't really want to admit it, unfortunately forgiving is almost never 100% truly complete. It's not like we can just act like nothing happened and even if we would that is called pretending that's not forgiving.
Outside, the words might be "I forgive you", but inside there's a different story.
And that will maybe become obvious the next time when the original offender does "something"; even if this time it's something really small -it's still might look a lot bigger because we tend to add things to the timeline. We always stitch experiences together, there is no delete memory function -if something happens and we have a working memory that is always there and will never be forgotten.
Now that we got that part out of the way, here's what can be done.
Keep in mind there is no guarantee. This is literally what can be done but it doesn't mean that you should expect anything specific to happen as a result. All we control is the input, not the output, hopefully, this makes sense. You can do whatever you want but the way the other person is going to respond that's entirely up to them and they should have this right.
What can be done?
First, you have to get another chance like this person should be willing to accept to spend more time with you, without the chance failure is guaranteed. So then the question becomes how do you get another chance? But you see a chance for what; because this is not a chance as in a new relationship but just another chance to demonstrate that you have changed and that you deserve another shot at a relationship.
This is just a chance so you can demonstrate that you regret your mistake, that you will not do it again, and that you are willing and able to change.
So how do you make that happen? Well before you go out and try to convince this person you have to first be sure that all of the above is true. Before you beg for another chance you have to look inside and see if this is actually the truth, because if it's not you're going to deceive this person again.
Second, if they give you another chance make sure that you will be honest every day, have and demonstrate integrity even and especially when it's hard. When you say something make sure to keep your promise, and when in doubt always favor the truth. Over time this new layer of trust is going to be built on top of the part that was broken and hopefully if you do a good job your relationship is going to become even stronger.
Ultimately trust is not something that you can buy from the market you can beg for it and get more of it trust is a rare commodity it's built over time and lost in seconds.