mail broken by 10.9.2 update

Hello there, Joerg.

Something you may want to make sure of is that the “Ask for Keychain password” option hasn't been accidentally selected for any Gmail related items in your System keychain in Keychain Access under Access Control. You may find the following Knowledge Base article helpful:

OS X Mavericks: Allow apps to access your keychain

http://support.apple.com/kb/PH13737

Thanks for reaching out to Apple Support Communities.

Cheers,

Pedro.

Try going to Applications/Utilities/Keychain Access and deleting your password for Mail.

You can also try running Keychain First Aid and resetting the original keychain.

Keychain Reset