I agree w/ cal -- it's hard to define the limits of a Civ. Certainly it's not defined by national borders... What's good for a nation may not be good for a Civilization, for example.
I'd say it's a combination of language family, religion and some other cultural factors.
Samuel Huntington believes there are eight major modern Civs, and all of the ancient Civs have been absorbed or evolved into them --
Chinese
Japanese
Sub-Saharan African
Russian/Slavic
Western
Latin American
Islamic
Indian/Hindu
Strictly speaking, Egyptian civilization is the oldest, possibly rivalled by Sumerian. Both are physically proximal to Mecca, the cultural center of Islamic civilization, and share it's contemporary cultural commonality -- Islam.
So in a roundabout way, Islamic civilaztion is the oldest... but the nature of the culture has changed so much over 6,000 years it's completely different than in ancient times.