Delete mailbox but keep archive

I have a mailbox for an email address (private domain) from fatcow which I have been retrieving on its own in Apple Mail. I recently archived most of it and started setting it up to be retrieved through my generic Gmail account, which I also get on my Apple mail in a separate mailbox.

My question now is, since I'm getting these private domain emails through an existing Gmail mailbox already, is there a way to delete the former mailbox but keep the archives on my Apple Mail? I'm afraid to lose the archives if I delete the mailbox because I might still need some of the conversation history.

Thanks in advance!