Processors and motherboards have to be compatible: have the same pin interface at a minimum. So there are two questions:
1) Is there a dual core processor that is compatible with your motherboard? If not, you need a new motherboard.
2) Do you have the confidence and steady hands to remove the heatsink / fan assembly, remove the old processor, install a new one, apply thermal paste, and reinstall the heatsink? Without damaging anything? If not, then you may consider not trying it.
That's a seven-year-old motherboard. You would have to go to a flea market to get a compatible processor. "Socket 939 was succeeded by Socket AM2 in May 2006," from trusty old Wikipedia. Here's one, but you are better off saving up and doing a new build. http://www.amazon.com/Opteron-Up-Model-175-939/dp/B000B...