[Exim] Need help configuring Exim

I've installed Exim 3.3 on my Debian GNU/Linux server for my department.
I'm using the default setup. Here's how our current email works for our
city gov't and what I want Exim to do for our department. Our city gov't
currently uses an ISP for our email needs. Each user has a POP account
with the ISP and retrieves their mail by connecting to the ISP through
the network. All users email addresses end with nngov.com. I have a
Debian GNU/Linux server running in our Personnel Department. I have
Exim, Procmail, Fetchmail and UW-Imap running on it. I have added all
the employees in my department to the server's user list.

Currently, I have myself and three other users retrieving their email on
the server. I am using Fetchmail to retrieve each users email from the
ISP and store it in /var/mail/username. Right now, every user still
connects to the ISP for sending mail via SMTP. I would like our server
to be the SMTP server. I configured my mail client to use our server for
SMTP. I sent an email to one of the other users and it sent the mail
locally without going up to the ISP. Perfect. This is what I want it to
do. Here's where the confusion starts.

The problem is how do I get it to send email on to someone in another
department? Or even someone not in the city gov't? The domain names are
the same in each dept. That is, if I'm working in I.T., my email address
is jself@???. If I'm working in Personnel, my email address is
still jself@???. How do I get Exim to realize that if the user is
not working for Personnel, then it needs to send the email up to the
ISP? I get an error message that says mailbox cannot be found when I
send an email to someone with the nngov.com domain but who is not a user
on my server. And if the email address I want to send email to is
something other than nngov.com, i.e. someone@???, then Exim
needs to send this email up to the ISP as well.

Sorry this is so long.
--
Jeff Self
Information Technology Analyst
City of Newport News
Personnel Department Suite 200
700 Town Center Drive
Newport News, VA 23606
757-926-1810