I have been in the same situation and I'm going to answer NO.
When one person is incapable (for any reason) of wanting to rebuild that trust, it's a fruitless effort.
I have been paying for my mistake for 2 1/2 years, and he refused to try to work on rebuilding trust. I have been in an absolute hell-hole filled with abuse. (not to say that you will go through the same) but if a person does not want to try to work on it, then there's nothing you can ever so or do to change that. I lied, and he told me that he shouldn't have to put in any effort or work since I'm the one that lied.
Now I can't get out of this relationship, he won't let me out and I will have to secretly move away.
It's terrible.
Apologize for your breaking the trust, and if the other person can't accept that and refuses to work on it with you, cut your losses and move on. It wasn't meant to be and you will find the person you were meant to be with. (as cliche as that may sound, but if this person cares to make it work with you, then they will do their part too)
Good luck and let me know if you need to chat. I'm in the same boat.