Woman who accepts Islam and whose husband is a kaafir
All praise be to Allah the Almighty.
If a woman becomes a Muslima and her husband is a kaafir, then it becomes mandatory for her to separate from him and it is not permissible for her to remain with him in his state of shirk (disbelief), as per Allah's injunction (interpretation of the meaning):
" do not send them back to the disbelieversthey are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them" (Al-Mumtahinah 60:10).
It is her duty to invite him to Islam and to attempt to convince him via various approaches in the hope that Allah may accept her efforts and give her the great reward for his guidance through her. The prophet (peace be upon him) once told Ali (may Allah be pleased with him) when he sent him to a people who were idol-worshippers:
"Invite them to Islam and inform them what is required of them, for (I swear) by Allah, that Allah guides a man through you is better than if you had humr un-na'am (a particular kind of camel which is the most sought-after wealth among the Arabs)" (Al-Bukhari, Fath il-Baari 3009).
As for the course of the marriage, and what she should do if her husband accepts Islam or doesn't, Ibn ul-Qayyim (may Allah have mercy upon him) said: What is indicated by evidence of his (i.e., the prophet's) ruling is that the marriage is mawqoof, or in abeyance (i.e. suspension); if her husband accepts Islam before the elapsing of her 'iddah (waiting period), then she remains his wife. If her 'iddah elapses then she has the right to marry whom she pleases or if she likes she may wait (in the hope he may still accept Islam). If he then accepts Islam, she becomes his wife without need of renewing the marriage. (Zaad ul-Ma'aad, Vol 5 p.138).
wallahu a'lam. (And Allah the Most Knowledgeable knows what is the most true and correct)