Is this your motherboard.?http://www.gigabyte.com/products/product-page-aspx?pid=2255#ov
Check the CPU support list.. It does not shows any supported C2D's.. However, it does has a pci-express X16 slot which makes any pci express video card compatible to it provided your PSU supports it.. An HD 5670 will be fine.. Just that you won't be able to enjoy the card to its fullest as your weak processor will severely hold it back..