We present a measurement of time-dependent CP-violating asymmetries in neutral B meson decays collected with the BABAR detector at the PEP-II asymmetric-energy e+e- collider at the Stanford Linear Accelerator Center. The data sample consists of 29.7 fb-1 recorded at the γ(4S) resonance and 3.9 fb-1 off resonance. One of the neutral B mesons, which are produced in pairs at the γ(4S), is fully reconstructed in the CP decay modes J/ ψKS0, ψ(2S)KS0, χc1KS0, J/ψK*0 (K*0→KS0π0) and J/ψKL0, or in flavor-eigenstate modes involving D(*)π/p/a1 and J/ψK*0 (K*0→K+π -). The flavor of the other neutral B meson is tagged at the time of its decay, mainly with the charge of identified leptons and kaons. A neural netsvork tagging algorithm is used to recover events without a clear lepton or kaon tag. The proper time elapsed between the decays is determined by measuring the distance between the decay vertices. Wrong-tag probabilities, the time-difference resolution function, and the B0-B̄0 oscillation frequency Δmd are measured with a sample of about 6350 fully-reconstructed B0 decays in hadronic flavor-eigenstate modes. A maximum-likelihood fit to this flavor eigenstate sample finds Δmd=0.516±0.016(stat)±0.010(syst) ps -1. The value of the asymmetry amplitude sin 2β is determined from a simultaneous maximum-likelihood fit to the time-difference distribution of the flavor-eigenstate sample and about 642 tagged B0 decays in the CP-eigenstate modes. We find sin 2β=0.59 ±0.14(stat)±0. 05(syst), demonstrating that CP violation exists in the neutral B meson system. We also determine the value of the CP violation parameter |λ|=0. 93±0.09(stat)±0.03(syst), which is consistent with the expectation of |λ|=1 for no direct CP violation. © 2002 The American Physical Society.