If you look at the post I made a bit earlier today, you'll see what I found the problem to be - Apple changed the DNS name of the IMAP & SMTP servers. Unfortunately, it appears that Mail.app doesn't know about the change either. (The same appears true of the iCloud preference pane as well).
So when Maii.app tries to connect to mail.me.com, it fails, because the IMAP server isn't there anymore.
So try this minor change to the Mail.app instructions given by Csound1:
Click + to add a new email account.
put in your name, and the first part of your email address (DO NOT put in @me.com @mac.com, or @anything.com)
Hold down Opt and then click "continue"
Now you'll have the manual setup window.
Set account type to IMAP
for the incoming mail server, use imap.mail.me.com (NOT mail.me.com)
After doing this, then Mail.app worked for me.
If I put in @me.com or @mac.com in the email address for example, Mail.app would crash & burn before i could even put in the IMAP server location. So it appears that if you try to keep mail.app from ever knowing you're trying to connect to iCloud (until you feed in imap.mail.me.com as the IMAP server), Mail.app will not try to be "smart" and use an invalid configuration.