We present an updated measurement of time-dependent CP-violating asymmetries in neutral B decays with the BABAR detector at the PEP-II asymmetric B Factory at SLAC. This result uses an additional sample of Υ(4S) decays collected in 2001, bringing the data available to 32×106 B¯B pairs. We select events in which one neutral B meson is fully reconstructed in a final state containing charmonium and the flavor of the other neutral B meson is determined from its decay products. The amplitude of the CP-violating asymmetry, which in the standard model is proportional to sin2β, is derived from the decay time distributions in such events. The result sin2β=0.59±0.14(stat)±0.05(syst) establishes CP violation in the B0 meson system. We also determine |λ|=0.93±0.09(stat)±0.03(syst), consistent with no direct CP violation.