Memtest86
http://www.memtest86.com/memtest34a.zipUnzip, run rawrite.exe and it will make a bootable floppy disk... boot the computer with the floppy and the memtest screen will appear... run the test and if the memory fails it will halt with a flashing red screen to let you know
if you dont have a floppy drive you can use a CD and burn it using this file
http://www.memtest86.com/memtest86-3.4a.iso.zipbut in order to make that CD you must have burn softwarer installed because the WindowsXP default burner software may not work unless you have the ISO update for it.
if you have 2 stick of memory instead of 1.. if it fails, remove one stick and run the test again... if after a few hours it passes without any failures, swap sticks and run the test again.. that way you can figure out which stick may be bad.
All that assumes the motherboard is OK and the problem is not some type of failure of the motherboard to power the memory. Rare, but also possible. usually a failure is a bad stick.