I use to subscribe the view that TCassidy is promoting, however I will state my current views now:
I believe the exception clause refers to
1. Incestuous marriages (See Leviticus 18) (I also believe all sodomite "marriages" and sad that I have to say in our time and age but beastiality "marriages" would also be invalid as well, though I realize our passage in question relates to husband putting away a wife)
2. Betrothal unfaithfulness
Matthew 1:19 Then Joseph her husband, being a just man, and not willing to make her a publick example, was minded to put her away privily.
Deuteronomoy 22:13-21
If any man take a wife, and go in unto her, and hate her,
And give occasions of speech against her, and bring up an evil name upon her, and say, I took this woman, and when I came to her, I found her not a maid:
Then shall the father of the damsel, and her mother, take and bring forth the tokens of the damsel's virginity unto the elders of the city in the gate:
And the damsel's father shall say unto the elders, I gave my daughter unto this man to wife, and he hateth her;
And, lo, he hath given occasions of speech against her,saying, I found not thy daughter a maid; and yet these are the tokens of my daughter's virginity. And they shall spread the cloth before the elders of the city.
And the elders of that city shall take that man and chastise him;
And they shall amerce him in an hundred shekels of silver, and give them unto the father of the damsel, because he hath brought up an evil name upon a virgin of Israel: and she shall be his wife; he may not put her away all his days.
But if this thing be true, and the tokens of virginity be not found for the damsel:
Then they shall bring out the damsel to the door of her father's house, and the men of her city shall stone her with stones that she die: because she hath wrought folly in Israel, to play the whore in her father's house: so shalt thou put evil
Some other things to consider:
1. The NT teaching for Husbands
Eph 5:25-28
Husbands, love your wives, even as Christ also loved the church, and gave himself for it;
That he might sanctify and cleanse it with the washing of water by the word,
That he might present it to himself a glorious church, not having spot, or wrinkle, or any such thing; but that it should be holy and without blemish.
If Husbands are to love their wives as Christ loved the church, how can we justify someone putting his wife away for adultery? Does Jesus divorce us and end his relationship with us when we are unfaithful to him?
2. Suffering for righteousness
1 Peter 2:20-21 For what glory is it, if, when ye be buffeted for your faults, ye shall take it patiently? but if, when ye do well, and suffer for it, ye take it patiently, this is acceptable with God.
For even hereunto were ye called: because Christ also suffered for us, leaving us an example, that ye should follow his steps:
3. God hates divorce
2:14-16 Yet ye say, Wherefore? Because the LORD hath been witness between thee and the wife of thy youth, against whom thou hast dealt treacherously: yet is she thy companion, and the wife of thy covenant.
And did not he make one? Yet had he the residue of the spirit. And wherefore one? That he might seek a godly seed. Therefore take heed to your spirit, and let none deal treacherously against the wife of his youth.
For the LORD, the God of Israel, saith that he hateth putting away: for one covereth violence with his garment, saith the LORD of hosts: therefore take heed to your spirit, that ye deal not treacherously.
Matthew 19:4-8 And he answered and said unto them, Have ye not read, that he which made them at the beginning made them male and female,
And said, For this cause shall a man leave father and mother, and shall cleave to his wife: and they twain shall be one flesh?
Wherefore they are no more twain, but one flesh. What therefore God hath joined together, let not man put asunder.
They say unto him, Why did Moses then command to give a writing of divorcement, and to put her away?
He saith unto them, Moses because of the hardness of your hearts suffered you to put away your wives: but from the beginning it was not so.
4. Other NT Passages
1 Corinthians 7:10-11 And unto the married I command, yet not I, but the Lord, Let not the wife depart from her husband: But and if she depart, let her remain unmarried, or be reconciled to her husband: and let not the husband put away his wife.
Romans 7:2-3
For the woman which hath an husband is bound by the law to her husband so long as he liveth; but if the husband be dead, she is loosed from the law of her husband.
So then if, while her husband liveth, she be married to another man, she shall be called an adulteress: but if her husband be dead, she is free from that law; so that she is no adulteress, though she be married to another man.
5. When you make a vow that "Till death do us part" I do not believe adultery gives you the right to break that covenant and promise you made to your spouse to love them and care for them "Till death do us part".
6. If you were committing adultery, would you want your spouse to divorce you? or would you want her to love you and pray for you until you came to a place of repentance? We are to do to others as we would have done unto ourselves.
Just some final notes, I do not see how the verse in Corinthians that says "is not under bondage" can mean they are free to remarry, in my opinion that is something read into the passage. Also I realize that under the Law of Moses God allowed remarriage, however, it is said that was for the hardness of their hearts, I do not think God ever was happy with any kind of divorce, God clearly hates divorce, and New Testament Christians should go far beyond the letter of the law of the Moses because we have the Spirit of God indwelling us.