the iCloud IMAP server "p01-imap.mail.me.com" rejected the password for user xxxxxx I reset my Apple ID (@me.com - Only have ONE account) as Apple asked me to update my information with a more strong password. I can login just fine into iCloud mail from the web and check, see my emails but as soon as I launch the Mail app this window keeps popping up over and over telling me that my password was rejected for user by the iCloud IMAP server. I have removed iCloud from my Mac and add it all back without any complains or issues but as soon as I launch the Mail application under my @me.com account this window pops-up. I don't know what else to do and so far I haven't been able to solved the problem. Any ideas? My machine is a 2011 MBP with OS Lion.

My Mail Application configurations under Incoming Mail Server shows: p01-imap-mail.me.com and Outgoing Mail Server (SMTP): iCloud (iCloud).
It should be in the format p01-imap.mail.me.com, it looks like your missing the 'mail' part.