Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
If the wife has sincerely repented from adultery, and the husband is convinced of it, then he is permitted to keep the marriage. However, if there is no sign of repentance, then it is not permitted for a Muslim to keep such a wife, or a husband as the case may be.
Responding to your question, Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
Adultery is one of the most heinous of sins in Islam. No Muslim is permitted to cohabit with a spouse who has committed such a heinous sin. Allah says, “the pure ones are for the pure.” and “the filthy ones for the filthy.”
Therefore, to cohabit with a person who is committing adultery, male or female, is sinful; it is, at once, repugnant to one’s Islamic sense and sound nature.
If the adultery of a spouse is proven or there is a confession, the spouses must immediately separate their beds. They are not permitted to relate to each other. However, divorce does not follow automatically. For divorce to take place, it must be pronounced or committed in writing.
If the wife has sincerely repented, and the husband is convinced of it, then he is permitted to keep the marriage. However, if there is no sign of repentance, then it is not permitted for a Muslim to keep such a wife, or a husband as the case may be.
Almighty Allah knows best.
Source: AsktheScholar.com