LinuxQuestions.org - Proper Exim + Sendmail configuration for outgoing emails. Howto?

Using dpkg-reconfigure exim4-config i setup Exim4 to use

Code:

internet site; mail is sent and received directly using SMTP

And get this kind of an error.
My DNS has a TXT record allowing this server/ip address to send emails from domain.com

Code:

2009-05-25 13:46:20 1M8eFg-0000Lm-G5 <= forum@domain.com U=www-data P=local S=375

2009-05-25 13:46:20 1M8eFg-0000Lm-G5 ** user@gmail.com: Unrouteable address

2009-05-25 13:46:20 1M8eFg-0000Lo-Go <= <> R=1M8eFg-0000Lm-G5 U=Debian-exim P=local S=1185

2009-05-25 13:46:20 1M8eFg-0000Lo-Go ** forum@domain.com: Unrouteable address

2009-05-25 13:46:20 1M8eFg-0000Lo-Go Frozen (delivery error message)

2009-05-25 13:46:20 1M8eFg-0000Lm-G5 Completed

I tried searching forum, but all answers doesnt seem to point me to anything.
Any idea?

Thank you.