If you have XP, then there is a utility in your accesories/system tools folder called files and settings transfer wizard. That will not only save your addressbook, but also your internet settings, favourites, and you can even make it save your files. It is self-explanatory.
If you have an earlier Windows, I believe the addressbook is stored in C:\Documents and Settings\Owner\Application Data\Microsoft if my memory serves me right. Just save it to a CD or floppy, and put it on your desktop on the new computer. Then Open outlook express and through the filetab you can navigete to import the addressbook.