Can my computer handle a processor upgrade?

I am graduating soon, and I will hopefully get money (hooray for me). Anyway, I was going to buy a new processor for my computer. I currently have a 1.8GHz single core, and I want to upgrade to at least a 2.0GHz Dual Core. My computer is really old, so I am not sure if it can handle it, or not. (It was made in 2005). Socket 939.