After booting Mandriva 2009 Free, my computer
hangs after the message: Determining IP
Information for eth0.

This is a wired nic- realtek RTL-8139/8139C/8139C+.
I don't have any problems connecting with it in 2008
Spring Free.

This is strange- I just tried the live cd and had no trouble
connecting.

So it's working fine on the LiveCD, just not with the full installation you did? How long does it stay there for? Normally, DHCP will wait for one minute before failing and continuing to boot.

It hung indefinately.

I tried various things-the last thing I tried was to boot
from the linux-nonfb kernel instead of the default one.
I was shocked that everything worked!!!

I have no idea why linux-nonfb worked and not
the default one.

What you could do is when booting, press the i for interactive mode, then you can choose what to load and not to load. We can then troubleshoot and see where the problem lies. Try booting without activating the network, and then we can see what happens.

What's the contents of your /boot/grub/menu.lst file? Can you post here and then we can see if there is anything obvious.