New flat screen from Dell.
 

Plug and Play on both my Fedora Core2 boxes, no issues. No display on my Fedora Core 1 box. Please help...

By "no display" I assume you mean no X server, correct? Still have the old monitor? Or a newer live cd? My first two thoughts are these two options:

First, I'd try a newer live cd. If it boots in the FC1 box with the monitor plugged in, I'd backup your /etc/X11/xorg.conf on your FC1 partition to /etc/X11/xorg.conf.backup or whatever you wish to call it and copy the one from /etc/X11/xorg.conf on the live cd to your FC1 partition and reboot without the disc to see if it worked.

Or, try this:

Hook the old monitor back up to the FC1 box and backup the /etc/X11/xorg.conf on your FC1 box to /etc/X11/xorg.conf.backup or whatever you want to call it and then copy the /etc/X11/xorg.conf from your FC2 box to your FC1 box and reboot. See if that fixes it.

My first two thoughts...