Praise be to Allah.
It is not permissible for a Muslim to remain in a marital relationship with a wife who is not Muslim, because Allaah says (interpretation of the meaning):
“And do not marry Al-Mushrikaat (idolatresses) till they believe (worship Allaah Alone) [al-Baqarah 2:221]
“Likewise hold not the disbelieving women as wives” [al-Mumtahanah 60:10]
“They are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them” [al-Mumtahanah 60:10]
‘Umar (may Allaah be pleased with him) divorced two of his wives who were mushriks when this aayah was revealed. Ibn Qudaamah stated in al-Mughni that there is ijmaa’ (scholarly consensus) on this point. He said, “There is no difference of opinion among the scholars that their women are forbidden (for marriage).” (al-Mughni 7/503)
Allaah has made an exception in the case of women from among the People of the Book [Jews and Christians]. He says (interpretation of the meaning):
“(Lawful to you in marriage) are chaste women from the believers and chaste women from those who were given the Scripture (Jews and Christians) before your time” [al-Maaa’idah 5:5]
The word “chaste” (muhsanaat) means that they do not engage in zinaa (unlawful sexual relationships).
As the wife mentioned in the question is not from among the People of the Book, her Muslim husband has to fear Allaah and leave her, because his relationship with her is haraam according to sharee’ah, and persisting in it is zinaa and is haraam. If a woman becomes Muslim and her husband is a kaafir, whether he is from among the People of the Book or not, the marriage contract immediately becomes null and void, because of the evidence cited above. She becomes forbidden to him by virtue of her becoming Muslim, and she is no longer permissible for him, unless he becomes Muslim during her ‘iddah (waiting period following divorce). And Allaah knows best.
See a similar question in Fataawa Islaamiyyah, compiled by Muhammad al-Musnad, 3/229.