Can you ping from the laptop?
Try bringing up a dos prompt on the desktop, and typing "ping news.bbc.co.uk". Make a note of the IP address.
Now, try the same command twice on the laptop. Once, use news.bbc.co.uk and the second time try the IP address. If the second one works but the first fails then it is a DNS problem.
Oh, and is the desktop machine firewalled...?