The Gigabyte website is a bit of a challenge on the CPU support list, but it looks like the Rev 4.1 or newer rev 5 of that board will work as of BIOS version F3. That is for the Vishera 8320 chip and not specifically for your 8310.
Even so, like crash pointed out, if it supports the chip FAMILY, it should support the OEM chip, as they tend to NOT test the OEM chips for board compatibility. I'd say that board, provided it is a rev 4.1 or 5 with the FE BIOS on it is as good as it will get.
A 4+1 phase power design means overclocking will be very limited, but you pointed out already that this is not a concern.