[Exim] Email on internal LAN

I have a machine on LAN that receives internet mail for domain y.com. I
have a second machine on same LAN that can receive mail at x.y.com. I want
to be able to send mail to y.com from x.y.com but exim is trying to connect
to the public ip address of y.com which it gets from internet DNS server.
This fails due to firewall/nat configuration. How can I send mail to the
machine using the LAN address of y.com? There is no DNS server on the LAN.

Thanks for any help.
Israel