Uhhm, give it the Windows ME disk?
Nah, only joking. I take it since ME is an old, old OS that the computer is second hand and you don't have the disk? Meh, I would say just get a different operating system. Windows ME is completely terrible in every way!
Linux would be a good choice, as a lot of Linux versions are designed to run on old computers (look-up Damn Small Linux, it can run on computers as old as 1994!). That said, if you're not willing to go down the Linux route, Windows 2000 and Windows XP are both pretty good, and should run reasonably on old hardware
EDIT: If you really don't want to update your computer, Damn Small Linux can actually run on top of Windows, so you would just boot into Windows as normal, and THEN fire-up Damn Small Linux! At least it would save you changing your OS 