  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 Upsilon(4S) decays collected
in 2001, bringing the data available to 32 million B-anti-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 sin2beta, is derived from the decay time
distributions in such events. The result sin2beta = 0.59 +/- 0.14 (stat) +/-
0.05 (syst) establishes CP violation in the B^0 meson system. We also determine
|lambda| = 0.93 +/- 0.09 {stat} +/- 0.03 {syst}, consistent with no direct CP
violation.
