Mail Downloads Duplicate Mail Repeatedly

I have had three accounts, each on its own domain, set to receive mail from godaddy.com's POP servers for several years.

Since yesterday Dec. 17th, Apple Mail has started downloading duplicates of all emails left on the POP server for one of my accounts, a fresh copy every time it checks.

This account is set in Mail to leave copies on the server for one month, as backup and so I can also retrieve messages on the web and with my iPhone and iPad.

I have tried rebuilding the mailbox, restarting my Mac and relaunching Mail, dumping the Envelope Index from User/Library/Mail and relaunching/rebuilding the Envelope Index. I have deleted the account from Apple Mail and set it up again from the top. I have changed Mail settings to delete messages immediately from the server, then reset it to leave messages on the server after they had all been cleared; new test messages persist in downloading duplicates with every message check.

I have tried turning off all Rules in preferences; I tried setting a rule to deal with duplicate messages, but there is no such option in the pop-up menus.

Mail on my iPad and iPhone do not appear to suffer from this problem, and can discern between messages already downloaded and new ones that appear.

My other two accounts were set to delete messages immediately; changing this setting as a test to leave messages on the server began to produce the same duplicate download problem.

I contacted support at godaddy.com; they looked into it and re-synced my account but found nothing wrong and say there has been no wave of Mac users reporting this problem to them.

*UPDATE* As a workaround, I have installed and migrated my files to Mozilla Thunderbird and stopped running Apple Mail, except kept offline and using only as sync master for iPad and iPhone Mail settings.