Praise be to Allah.
It is permissible to marry a Christian woman, if she is chaste and refrains from zina (unlawful sex), because Allah, may He be exalted, says (interpretation of the meaning):
“This day [all] good foods have been made lawful, and the food of those who were given the Scripture is lawful for you and your food is lawful for them. And [lawful in marriage are] chaste women from among the believers and chaste women from among those who were given the Scripture before you, when you have given them their due compensation, desiring chastity, not unlawful sexual intercourse or taking [secret] lovers. And whoever denies the faith - his work has become worthless, and he, in the Hereafter, will be among the losers”
But her Muslim husband must have guardianship over her and his children from her, because Allah, may He be exalted, says (interpretation of the meaning):
“and never will Allah give the disbelievers over the believers a way [to overcome them]”
Ibn Jareer (may Allah have mercy on him) pointed out an important condition having to do with this, which is that it should be in a country where the man who marries her will not fear that his child may be compelled to disbelieve.
End quote from Tafseer at-Tabari (9/589).
If he fears that the law of the land in which he lives will overrule his guardianship over his wife and family, or waive his right to custody of his children, or will give his Christian wife the right to make his children Christian, then if he is able to stipulate his rights and take precautions for himself and his progeny (then in that case he may marry her), otherwise it is not permissible for him to go ahead with that marriage.
It is better to marry a Muslim woman, because she is more deserving of his marrying her and keeping her chaste, and spending on her, and it is usually more beneficial for the child and his upbringing, because if he marries a woman of the people of the Book, there is the fear that she may corrupt her children or make them follow her religion.
It says in Kashshaaf al-Qinaa‘ (5/84): It is more appropriate not to marry their women. The shaykh said that is makrooh (disliked).
That is when there are free Muslim women. It says in al-Ikhtiyaaraat: This was stated by al-Qaadi and most of the scholars, because ‘Umar said to those who married women from the People of the Book: Divorce them. End quote.
The guardian of a Christian woman should be someone who follows her religion among her male relatives. As for an atheist, he is not qualified to be her guardian, because of the difference of religion.
It says in Kashshaaf al-Qinaa‘ (5/35), explaining the conditions to be met by the guardian: … The third is that they should be of the same religion, both the guardian and the woman whose guardian he is. So a disbelieving man should not act as a guardian for a Muslim woman when she gets married, or vice versa. It says in al-Ikhtiyaaraat: (Similarly, it is not valid) if the woman is Jewish and her guardian is Christian, or vice versa. Something similar was stated in Sharh al-Muntaha, where it says: A Christian man has no right of guardianship over a Magian (Zoroastrian) woman, and so on, because there is no right of inheritance on the basis of blood ties between them. … The ruler (or judge) act as guardian for a disbelieving woman who has no guardian. End quote.
Based on that, if there is no Christian man among her male relatives, then the Muslim judge should act as her guardian in her marriage, if there is one. If there is no Muslim judge, then the director of the Islamic Centre in her area should act as her guardian in her marriage.
It says in Fataawa al-Lajnah ad-Daa’imah (18/322): It is not permissible for a Muslim man to marry a Christian woman unless she is chaste and refrains from zina (unlawful sex), and her marriage contract should be done by her guardian, namely her father. If he is not present, then it should be done by the closest of her male relatives, because the Prophet (blessings and peace of Allah be upon him) said: “There can be no marriage without a guardian.”
If she has no guardian, then the mufti of the Muslims, or the director of the Islamic Centre in your city, may act as her guardian in marriage. It is not permissible for her mother to act as her guardian in marriage, because the mother has no right of guardian over her in the issue of marriage. End quote.
And Allah knows best.