A Muslim woman is NOT permitted by Islam to marry a non-Muslim. This is an established fact in Islam that is as clear as the daylight.

Having said this, we should stress here that if this man willingly accepts Islam out of pure conviction and not just to get married, then a lady is permitted as a Muslimah to marry him.

In this regard, we would like to cite the following issued by the eminent Muslim scholar, Sheik Yusuf Al-Qaradawi, in his well-known book, The Lawful and the Prohibited in Islam:
“It is haram for a Muslim woman to marry a non-Muslim man, regardless of whether he is of the People of the Book or not. We have already mentioned the saying of Allah, “…and do not marry (your girls) to idolaters until they believe…” (Al-Baqarah: 221)
And He said concerning the immigrant Muslim women: “Then if you know them to be Believers, do not send them back to the unbelievers. They are not lawful for them (as wives), nor are they lawful for them (as husbands).” (Al-Mumtahanah: 10)
No text exists which makes exceptions for the People of the Book. Hence, on the basis of the above verses, there is a consensus among Muslims concerning this prohibition.
Thus, while a Muslim man is permitted to marry a Christian or Jewish woman, a Muslim woman is not allowed to marry a Christian or Jewish man. There are many sound reasons for this difference. First, the man is the head of the household, the one who maintains the family, and he is responsible for his wife. And while Islam guarantees freedom of belief and practice to the Christian or Jewish wife of a Muslim, safeguarding her rights according to her own faith, other religions, such as Judaism and Christianity, do not guarantee the wife of a different faith freedom of belief and practice, nor do they safeguard her rights. Since this is the case, how can Islam take chances on the future of its daughters by giving them into the hands of people who neither honor their religion nor are concerned to protect their rights?
a marriage between a man and woman of different faiths can be based only on the husband’s respect for his wife’s beliefs; otherwise a good relationship can never develop. Now, the Muslim believes that both Judaism and Christianity originated in divine revelation, although later distortions were introduced into them. He also believes that God revealed the Tawrah to Moses and the Injeel to Jesus, and that both Moses and Jesus (peace be on them) were among the Messengers of Allah who were distinguished by their steadfast determination. Accordingly, the Christian or Jewish wife of a Muslim lives under the protection of a man who respects the basic tenets of her faith, her Scripture, and her Prophets, while in contrast to this the Jew or Christian recognizes neither the Divine origin of Islam, its Book, or its Prophet (peace be on him). How then could a Muslim woman live with such a man, while her religion requires of her the observance of certain worships, duties, and obligations, as well as certain prohibitions. It would be impossible for the Muslim woman to retain her respect for her beliefs as well as to practice her religion properly if she were opposed in this regard by the master of the house at every step.
It will be realized from this that Islam is consistent with itself in prohibiting the Muslim man to marry a mushrik (polytheist) woman, for since Islam is absolutely opposed to shirk (polytheism), it would obviously be impossible for two such people to live together in harmony and love.”