Unless there is an extremely good reason for the divorce, then divorce and remarriage is adultery. God doesn't recognizance the new marriage, he sees it as an adulterous relationship, as Christ made expressly clear.
I dread this topic, because it is the source of so much suffering in the world, and my immediate inclination is simply to respond to people with compassion rather than by reciting laws. However this is a question about God's law and so I am going to stick to the point.
Matthew 19:9 doesn't make any exceptions for "extremely good reasons," I'm afraid. Remarrying after a divorce is adultery, period.
HOWEVER, and this is really important, there are such things as invalid marriages, relationships that appear to be marriages but which are illicit for any number of reasons. Perhaps there was fraud involved, or force, or the inability to give consent due to mental illness or intoxication. Sometimes the necessary criteria of a marriage simply doesn't exist, as in a gay marriage or an incestuous marriage. Sometimes there was no agreement to remain faithful or to otherwise fulfill the obligations of a marriage. The thing is, these invalid marriages almost always end up in civil divorce. What does the Bible have to say about this?
Matthew 19:9 says, I say to you, whoever divorces his wife EXCEPT FOR PORNEIA and marries another commits adultery.
Porneia is a Greek word that means illicit sexual intercourse (such as what is listed in Leviticus 18). The pagans in Jesus' time had so-called marraiges based on illicit relationships such as incest, relationships that the Jews would only call sinful. What Jesus was was saying in Matthew 19:9 is that a person could and should leave these kind of "marriages" and remarry and it wasn't adultery since they were invalid.
Thus, a really good translation for Matthew 19:9 would be the one in the New American Bible:
I say to you, whoever divorces his wife (unless the marriage is unlawful) and marries another commits adultery.
Here is a link to the grounds for which the Catholic church considers a marriage invalid:
http://www.stmarys-waco.org/documents/Grounds for Marriage Annulment in the Catholic Church.pdf