[Exim] Domain name rewriting from local senders to non-local…

Dear all

Our school network uses a different internal domain to that by which we are
known on the Internet.

Both domains are set in the configure file to be recognised as local
domains, but when sending from a local client on our network (OE5 on
Windows), the local domain name is passed on (naturally enough) to our ISPs
upstream smtp server. The message is refused as the internal domain is not
recognised.

Where and how should I configure the rewriting of the domain from local
senders to non-local recipients (or any recipient, for that matter)?

Setting the "Reply To" option in MS OE5 doesn't seem to do it.

Perhaps I should re-configure our local domain to match our public name? Or
register our local domain
as an alias to our public domain?

I have made a concerted effort to understand all the documentation I have
read to date, promise!

Thanks in advance.
Martyn Hill
Network Administrator
St James Independent School
London