Praise be to Allah.
What you have said in your question about it being haraam (unlawful)for a Muslim woman to marry a non-Muslim man is correct, and there is no doubt concerning that.
Allaah says (interpretation of the meaning):
And give not (your daughters) in marriage to Al-Mushrikoon (polytheists)till they believe [al-Baqarah 2:221]
And give not (your daughters) in marriage means, do not give Muslim women in marriage to Mushrik (pagan)men. The ummah (Muslim nation)is agreed that a Mushrik should not marry a Muslim woman because this is like putting Islam in an inferior position. (Tafseer al-Qurtubi, 3/72).
Allaah says (interpretation of the meaning):
They are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them [al-Mumtahanah 60:10]
Al-Bukhaari (may Allaah have mercy on him) said:
Chapter: when a mushrik or Christian woman who is married to a dhimmi (non-Muslim living under Muslim rule) or a harbi (non-Muslim belonging to a people who are hostile towards Islam) becomes Muslim. Abd al-Waarith said, narrating from Khaalid from Ikrimah from Ibn Abbaas: if a Christian woman becomes Muslim shortly before her husband, she is forbidden for him Mujaahid said: if he becomes Muslim during the iddah [waiting period following divorce], then he may (re)marry her. Allaah says (interpretation of the meaning): They are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them [al-Mumtahanah 60:10]. Al-Hasan and Qutaadah said concerning two Magians who became Muslim that their marriage was still valid. If one of them had become Muslim and the other had refused, the woman would have been divorced and he would no longer have any rights over her.
(Saheeh al-Bukhaari. See al-Fath, 9/421).
Examples of such women include:
Zaynab, the daughter of the Prophet (peace and blessings of Allaah be upon him). She was married to Abul-Aas ibn al-Rabee during the Jaahiliyyah (pre-Islamic days of ignorance) but when she became Muslim, their marriage was annulled, and she went and stayed with her father (peace and blessings of Allaah be upon him). When her husband became Muslim, the Prophet (peace and blessings of Allaah be upon him) sent her back to him.
(Narrated by al-Tirmidhi, 1143; Abu Dawood, 2240; Ibn Maajah, 2009; classed as saheeh by Imaam Ahmad, 1789. Al-Tirmidhi said, there is nothing wrong with its isnaad).
The correct view is that the husband can go back to her with no need to renew the marriage contract.
If the woman is still in her iddah (waiting period), he has more right (than anyone else) to marry her. If her iddah has ended, she is free to choose whether to go back to him or not.
On the basis of this hadeeth, the scholars said that if a woman becomes Muslim before her husband, then her husband becomes Muslim whilst she is still in her iddah, then the husband has more right to her whilst she is still in her iddah. This is the view of Maalik ibn Anas, al-Oozaai, al-Shaafai, Ahmad and Ishaaq.
(Sunan al-Tirmidhi, Hadeeth 1142).
Ibn Abd al-Barr said:
There is no dispute among the scholars concerning the fact that if a non-Muslim woman becomes Muslim then her iddah ends, her husband has no rights concerning her if he has not become Muslim during her iddah.
Ibn al-Qayyim said:
But what the ruling of the Prophet (peace and blessings of Allaah be upon him) indicates is that the marriage comes to a halt. If he becomes Muslim before the end of her iddah, then she is (still) his wife, but if her iddah ends, then she may marry whomever she wants. If she likes, she can wait for him, and if he becomes Muslim she is his wife and there is no need to renew the marriage contract.
(Zaad al-Maaad, 5/137, 138)
Talhah ibn Ubayd-Allaah was married to Arwaa bint Rabeeah ibn al-Haarith ibn Abd al-Muttalib. They were separated by Islam, then in Islam Khaalid ibn Saeed ibn al-Aas married her. She was one of the [Muslim] wives of the non-Muslims who fled to the Prophet (peace and blessings of Allaah be upon him). The Prophet (peace and blessings of Allaah be upon him) kept her in Madinah and married her to Khaalid.
(Tafseer al-Qurtubi, 18/65, 66)
It was reported that Anas said: Abu Talhah married Umm Sulaym and the mahr (dowry)between them was Islam. Umm Sulaym became Muslim before Abu Talhah. He proposed marriage to her and she said, I have become Muslim. If you become Muslim I will marry you. So he became Muslim and that was the mahr between them.
(Narrated by al-Nasaai, 3340)
The daughter of al-Waleed ibn al-Magheerah, the wife of Safwaan ibn Umayyah, became Muslim before him, and the marriage was annulled. Then he became Muslim later on, and she went back to him. It was narrated by Maalik in al-Muwatta, 1132. Ibn Abd al-Barr said: I do not know of any unbroken saheeh isnaad for this hadeeth, but it is famous and well known to the scholars of seerah, Ibn Shihaab, the leader of the scholars of seerah, and al-Shubi. The fame of this hadeeth is stronger than its isnaad in sha Allaah.
Umm Hakeem bint al-Haarith ibn Hishaam, the wife of Ikrimah ibn Abi Jahl. Her marriage was annulled, then he became Muslim during her iddah, so she went back to her husband.
(Narrated by Ibn Abi Shaybah in al-Musannaf, 4/107)
And Allaah knows best.