put the laptop hdd into an usb external drive case and connect it to your desktop.

Mount the drive in an external, USB 2 shell (make sure it's powered or that you have a powered USB hub). Connect it to the newer system and try reading it. You may have to take ownership of the files if it's an NTFS-based disc.

If you want to access it from another computer while its still in the machine, set it up to be shared across your network. Otherwise, I'd go along with what the rest are saying and buy a USB cage.

Many thanks for the advice.  I've now bought a shell on ebay and been able to access the files on my old hard disk without any problems.  I guess my original question appeared very naive to those of you who know what you're doing, but prior to asking it on this board, I had contacted Dell's technical support chat service and been told that it would be impossible to access my old disk (!!).  Which all goes to show where the best advice can be obtained.