The Catholic Church has a similar belief. You have to get an annulment.
It basically boils down to this. When you get married you are making a promise for life. That's why you say, "until death do us part." Since the church doesn't recognize a legal divorce, you can't get married of be intimate with anyone else. The Church would see it as cheating since you are considered still married to the first person.
I don't know all the rules on annulments, but I do know that it's difficult to get one in the Catholic Church.
Now, with that said. Jesus died for our sins. We are sinners and God still loves us even though we sin. The Church may have a rule, but I personly don't think God would be so noneunderstanding as to turn away from a good person of faith if they tried to stay married but couldn't make it work.
