Well it is a way, but you'll have to know what you're doing and most probably re-compile the kernel for SCSI/RAID/NICs drivers.
It goes like this:
- boot new server with a linux live cd, create partition and format them;
- put the 250GB IDE hdd in the new server, boot with the live cd again, mount the old hdd, copy everything over to the new hdd, "chroot" to the new hdd (where you copied everything),run your bootloader (like "lilo"), remove old hdd, pray & boot
This is a mere proof of concept. Does it work? Yes, I did it myself. Does it require advanced Linux knowledge? Yes. Do I recommend it? Yes, if you know what you are doing.
The alternative would be a fresh install on the new server (latest distro,software etc) and copy users,home directories.configuration files,databases over. Whatever you do, DO NOT throw/sell/give the old 250GB HDD away, keep it for 6 - 12 months it might prove to be a lifesaver