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A number of users who have chosen to set up iCloud mail manually in the Mail application seem to encounter problems with continually being asked to sign in. This may well be a result of incorrect settings such as not setting 'Apple Token' for authentication under the accounts advanced settings.
A more general approach to remedying such problems would be to turn off Mail in System Preferences/Settings > iCloud, checking the account has been removed from Mail (and deleting it if it hasn't), restarting and then enabling iCloud Mail again from System Preferences/Settings > iCloud. This will ensure the correct settings are used.
Message was edited by: Winston Churchill