Apple Mail on iMac keeps asking for password

The Mail app on the Mac keeps asking for the password on one of my accounts. When I put the password in it states that the account already exists (which I know it does!!). It still won't connect to the mail server. In exasperation I have deleted the email account and tried to add it back. But it still asks for the password and tries to verify the account and then tells me the account already exists. I'm in an endless loop of insanity!

If anyone can help I'd appreciate it.

Thank you.