That's not difficult at all. Shut down your PBX. Install the gigabit network card (preferably an Intel pcie - they have better flow control for low latency traffic like video streaming). Keep your original mobo network connected and active (do not turn it off from the bios etc). Bring it back to life. After rebooting access your PBX through Webmin. In the network section you will see that now you have a second ethernet port available. Do whatever configuration you need to do through there, like static ip settings, netmasks, gateway, dns, or simply dhcp if that's how you have your net configured.
As you configure your second interface it will have a different ip address than the first interface but it will be in the same subnet, the same gateway, the same dns etc. Test it and make sure that it works - computer networking-wise (don't worry about the pbx for now). You should be able to ping in and out of the new interface. You should be able to run iperf in client or server mode and check that you are hitting around 950-970 mbps.
Once you have the new port fully configured and checked you can take the mobo interface out of service and assign its ip address to the new interface so that you do not need to change all your phones pbs registration address.
If you are on dhcp it's a bit easier. But with static IPs it's still quite easy.
One thing you should never do; never assign the mobo ethernet port's MAC address to the new ethernet card. The idea of spoofing the mac address is attractive to make life easier with dhcp management, but with most motherboards the ethernet controller on the motherboard is physical link layer active (for wake on lan, offline maintenance etc). So it would cause you hard to troubleshoot issues not withstanding the headache.
Phone System 