Is a Kohen allowed to marry a woman who is the daughter of a Jewish woman and non-Jewish man?
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According to Shulchan Aruch, Even Ha'Ezer 7:17 and commentaries there, they should not marry. However, if they did so, they may remain together and need not divorce.
This issue is disputed both in the Gemara and in the Rishonim. The ruling of the commentators of the Shulchan Aruch is that lechatchila a kohen should not marry her, but if he did marry her he is not required to divorce her. There have been cases where authorities have been lenient that even if a kohen was only engaged to her, he may marry her.