Okay, an update. In short, I'm sitting in front of my main machine writing this. However, I'm using a wired connection between it and the box with the modem. Following the help that I received in this thread, it took all of about two minutes to get it all working properly.
I think that the problem must have been the wireless router not wanting to route traffic internally - I could ping correctly, but as soon as it came to connecting up to the outside world, it wanted to route through its 'Internet port' - which wouldn't talk with the box with the modem.
Anyway, I'm now set to continue using this setup until my broadband is re-connected in about a week :P Thanks again for your help michaelk and fotoguy - it's most appreciated