My mother did just that, and she thought about remarriage, but remained single and celibate. She took Paul's writings on the topic seriously. When the husband grants his wife the Bill of Divorce, Jesus taught that except in the case of porneia, he causes her to commit adultery. So marrying the divorced woman, is biblically considered adultery, but the responsibility for the adultery occurring appears to be placed on the husband who granted her the Bill of Divorce. I don't see any difference in putting away with or without this Bill of Divorce, because this is contrasted with the first part of Matthew 5:31, where Jesus said "You have heard it was said that whoever puts away his wife must give her a Bill of Divorce, but I say unto you...". This whole argument that Jesus was saying that the man can put away his wife as long as he gives her the Bill of Divorce, seems to ignore the first part of that verse!