Er...Mesopotamia is a geogrphical area, not a civilisation (it would thus be equivalent to Europeans, rather than Romans, or Celts).
Pretty hard topic though. Depends on your definition of civilisaiton. I'm going to take a cop out, and say that because this is a history forum and implicilty relies on some form of record keeping, it's going to be the first group of people to use writing. By that definition, however, it's likely we don't know about the world's first civilisation, because another group which had spent more time making spears than devising advanced methods of communication killed them and destroyed all their writings. The oldest one I do know of, however, would be the Babylonians.