become the root user, run system-config-network,
on the Devices tab, highlight the eth0 interface and press the Edit button.
That should bring up the Ethernet Device panel and on Hardware Device tab
check the "Bind to MAC address" box and hit the probe button.
It should look at the ethernet hardware it can see which should now be the
new onboard one.
save the change with File ->Save and Quit
The tool has deactivate and activate buttons but I like to do the full restart with:
on the command line, that's assuming you use network and not NetworkManager,
if the latter do:
service NetworkManager restart
PS. if you like dealing with the raw config files, you can go to /etc/sysconfig/network-scripts and
move the ifcfg-eth1 to ifcfg-eth0 (that is "clobber" the old ifcfg-eth0 with the new ifcfg-eth1 file):
mv -f ifcfg-eth1 ifcfg-eth0
then do the respective restart like above.