We are planning on migrating from Sharepoint Portal Server 2001 to
Sharepoint Portal Server 2003 within the next month.
All the articles I am finding only explain the steps you take if you
upgrade the current server to 2003.

I need to find out how to migrate 2001 over to 2003 when they are on
separate servers. Our new server just arrived today and we'd like to
install Sharepoint 2003 on it soon while the old Sharepoint 2001 site is
still running for users on it's original server. How do I migrate over
2001 to the new server when we are ready for that document library?

Can I run the upgrade.exe on the old server and move the contents to the
new server?