Praise be to Allah.
The marriage of a Muslim woman to a kaafir, no matter what his religion, is invalid according to sharee’ah and their intimacy is tantamount to fornication. They must be separated, because Allaah says (interpretation of the meaning):
“And do not marry Al-Mushrikaat (idolatresses) till they believe (worship Allaah Alone). And indeed a slave woman who believes is better than a (free) Mushrikah (idolatress), even though she pleases you. And give not (your daughters) in marriage to Al-Mushrikoon till they believe (in Allaah Alone) and verily, a believing slave is better than a (free) Mushrik (idolater), even though he pleases you. Those (Al-Mushrikoon) invite you to the Fire, but Allaah invites (you) to Paradise and forgiveness by His Leave”
And Allaah says, stating that the Muslim women are not permissible in marriage for kaafir men (interpretation of the meaning):
“They are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them”
Al-Qurtubi (may Allaah have mercy on him) said in his book al-Jaami’ li Ahkaam al-Qur’aan (3/72): The ummah is unanimously agreed that a kaafir man cannot marry a Muslim woman at all because that undermines the position of Islam.
But if the man becomes a Muslim – as mentioned in the question – then he must re-marry the woman with a new marriage contract, because the first marriage contract was invalid and does not count for anything in sharee’ah.
Shaykh ‘Atiyah Muhammad Saalim said in his completion of the book Adwa’ al-Bayaan, 8/164-165:
Why is it permissible for a Muslim man to marry a kaafir woman from among the People of the Book (i.e., Jews and Christians), but it is not permissible for a Muslim woman to marry a kaafir man from among the People of the Book?
This question may be answered from two angles:
1 – That Islam should prevail and not be prevailed over. The role of maintainer and protector in marriage belongs exclusively to the husband, so the man may influence his wife and she may not be able to practice her religion as she should, and she may leave her religion altogether. Similarly the children will follow the religion of their father.
2 – Islam is comprehensive and other religions are limited, on which there is based a social matter that affects family life and the relationship between the spouses. This means is that if a Muslim man marries a Jewish or Christian woman, he believes in her Book and her Prophet, so he will deal with her on a basis of respect for her religion because he believes in it in general terms, and that will give them some common ground which may lead to her becoming Muslim as her own Book tells her. But if a Jewish or Christian man marries a Muslim woman, he does not believe in her religion so he will not have any respect towards her religion or her principles. There is no room for common ground with him with regard to something that he does not believe in at all. So there is no room for harmony or mutual understanding; there is no goodness in such a marriage, so it is forbidden in the first place.
So they must make a new marriage contract. And Allaah knows best.
With regard to the question about al-Salaah al-Naariyyah, please see question no. 7505.