Good News... Looking at the link provided by Tazz (thanks!) it looks like you should be able to swap these motherboards. They both use the Intel B440BX chipset. This was an extremely popular chipset, and easy to upgrade. Usually a new CPU is detected automatically (but it depends on your BIOS). It would be a good idea to download and install the latest BIOS from Gateway before removing the motherboard.
The only difference I notice between these motherboards is the onboard graphics. The G6-450 had 16MB of memory, while the G6-350 only had 8MB. Don't plan on playing any games, or running at high resolutions unless you install a different video card.
Bad News... Swapping motherboards does require a decent knowledge of computer components (identification), and there are bound to be a couple snags along the way. If you think you're up to the challenge, I can direct you to a couple sites that may help.