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 Υ(4S) resonance and 3.9 fb -1 off resonance. One of the neutral B mesons, which are produced in pairs at the Υ(4S), is fully reconstructed in the CP decay modes J/ΨK S 0 , Ψ(2S)K S 0 , X c1 K S 0 , J/ΨK* 0 (K* 0 →K s 0 π 0 ) and J/ΨK L 0 , or in flavor-eigenstate modes involving D(*)π/ρ/a 1 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 network 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 B 0 -B̄ 0 oscillation frequency Δm d are measured with a sample of about 6350 fully-reconstructed B 0 decays in hadronic flavor-eigenstate modes. A maximum-likelihood fit to this flavor eigenstate sample finds Δm d =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 B 0 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.