If a Kohan was married to a giyuerit by an observant ("orthodox") rabbi, and if the rabbi wasn't aware that he was marrying a Kohan and not an Israelite, is the marriage legitimate?
Sources requested as well.
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The marriage is valid regardless of the knowledge of the rabbi had at the time. The prohibition for a kohen to marry a giyores is only a Lav- negative commandment and thus would not invalidate the marriage the same way a brother and a sister marrying would, for that is a forbidden relationship and the children would be mamzerim. Rather the children of this relationship would be considered Chalalim- disqualified kohanim and they may not marry kohanim and they and their father may not Duchan. (Shulchan Aruch Even Haezer 6:8)
It is of course still prohibited to be married in the first place and after the fact to remain together, and every time they are intimate they are transgressing a serious prohibition.
For more on this prohibition, see this Aish.com article:
There is another issue, however. It is important to check if the "Kohen" is a real "Kohen." How reliable is the Kohanic tradition in the family? Just because someone's last name is "Cohen" does not mean that he necessarily has the status of a Kohen. To be considered a Kohen, one must have an unbroken tradition, as well as other factors too numerous to mention here. (Nevertheless, most people who have the name Cohen also have the status as Kohen.)
Rav Joseph B. Soloveitchik in a powerful address to the RCA rabbinical convetion in 1973 told the story of a couple (link):
The Rav told the story of a young man and woman who sought his assistance. She was a convert who later fell in love with this young man, whose increased interest in Judaism she sparked. The two became engaged and he visited his grandfather’s grave, where he discovered that he is a kohen. What could the Rav do? A kohen may not marry a convert and therefore, tragically, this couple could not wed. However, we must unhesitatingly surrender to the will of the Almighty. With sadness in his heart, the Rav shared in the suffering of this woman who had to lose the beloved man she helped bring back into the fold. She valiantly walked away from him, surrendering to the Almighty’s will.