When a person embraces Islam, he or she has to change his or her way of life and fully adapt himself to the teachings of Islam. Sometimes past mistakes have consequences that one cannot easily get rid of. However, even in these cases, a Muslim has to seek the Islamic remedy for his or her situation as there is no excuse for leading a sinful life under Islam.
Answering this question, the Permanent Fatwa Committee of the Assembly of Muslim Jurists of America(AMJA), stated,
A new Muslim should do his or her best to keep away from what Allah prohibits. The one who tastes the sweetness of faith will always hate to go back to kufr (disbelief) just as he or she hates to be thrown into Hellfire. It is good that you realize that Islam does not permit sexual relationships outside marriage. Almighty Allah describes the believers as (Those who guard their private parts, save from their wives or those whom their right hands possess, for then they surely are not blamable) (Al-Mu’minun 23: 5–6).
However, if one of the spouses remains in his or her religion and the conversion to Islam is not easy, it it is the wife, then the man who is a muslim can marry her after making sure that she has repented committing adultery. Allah allows Muslims to marry Christian or Jewish women who are chaste. Hopefully, this might guide her to embrace Islam. At any rate, the husband should keep on striving to call her to Islam under all circumstances. However the relationship with her outside lawful marriage is prohibited, even under the circumstances that she is a Christian as mentioned.