Thanks people.
It seems as though the new computer will have an ethernet port though I'm pretty certain the old one doesn't. The new one will be in a room without a telephone line so that will be the one that has the wireless connection with the ADSL router sitting next to the old one.
Just one point - does the ADSL router allow the two machines to talk to each other or does it simply split the internet connection?
Once all the hardware is up and running, what happens to the e mails? Will they be downloaded to both PCs or is it possible to send specific messages to specific inboxes? Sorry to sound a bit thick in these matters.