The nature of the first generation of stars in the universe remains largely unknown. Observations imply the existence of massive primordial stars early in the history of the universe, and the standard theory for the growth of cosmic structure predicts that structures grow hierarchically through gravitational instability. We have developed an ab initio computer simulation of the formation of primordial stars that follows the relevant atomic and molecular processes in a primordial gas in an expanding universe. The results show that primeval density fluctuations left over from the Big Bang can drive the formation of a tiny protostar with a mass 1% that of the Sun. The protostar is a seed for the subsequent formation of a massive primordial star.