[exim] Mail sent to root@FQDN doesn't get forwarded

Hi, I have a Ubuntu 12.04 LTS system, running Exim 4.76-3ubuntu3.1. The
system has mail configured as a smarthost, so it sends via my ISP's SMTP
server. Exim listens on localhost only (and the firewall is on). I have
a gmail address set up as a alias for root in /etc/aliases. If I send
mail to root or root@localhost, then it's forwarded to my external email
account correctly, via the smarthost. However if I send to root@FQDN,
the exim logs show that it's not forwarded to my gmail account - it
shows that it's sending it to root@FQDN. Then the email just vanishes -
I assume it's sent via the smarthost, which tries to deliver it back to
this machine, which fails because Exim is only listening on localhost.

The problem is that anything run via cron (cron-apt etc) mails the
output to root@FQDN, so the emails from all my cron jobs are vanishing.
Why is the forwarding email address ignored if mail is sent to
root@FQDN, but applied correctly if it's sent to root@localhost? I've
checked my hosts file & the FQDN resolves to 127.0.1.1. I tried changing
this to 127.0.0.1, but it didn't make any difference.....

Rich