Study of time-dependent<i>CP</i>-violating asymmetries and flavor oscillations in neutral<i>B</i>decays at the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Υ</mml:mi><mml:mn/><mml:mo>(</mml:mo><mml:mn>4</mml:mn><mml:mi>S</mml:mi><mml:mo>)</mml:mo><mml:mn/></mml:math>

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 ${e}^{+}{e}^{\ensuremath{-}}$ collider at the Stanford Linear Accelerator Center. The data sample consists of 29.7 ${\mathrm{fb}}^{\mathrm{\ensuremath{-}}1}$ recorded at the $\ensuremath{\Upsilon}(4S)$ resonance and 3.9 ${\mathrm{fb}}^{\mathrm{\ensuremath{-}}1}$ off resonance. One of the neutral B mesons, which are produced in pairs at the $\ensuremath{\Upsilon}(4S),$ is fully reconstructed in the CP decay modes $J/\ensuremath{\psi}{K}_{S}^{0},$ $\ensuremath{\psi}{(2S)K}_{S}^{0},$ ${\ensuremath{\chi}}_{c1}{K}_{S}^{0},$ $J/\ensuremath{\psi}{K}^{*0}$ ${(K}^{*0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{0})$ and $J/\ensuremath{\psi}{K}_{L}^{0},$ or in flavor-eigenstate modes involving ${D}^{(*)}\ensuremath{\pi}/\ensuremath{\rho}{/a}_{1}$ and $J/\ensuremath{\psi}{K}^{*0}$ ${(K}^{*0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}).$ 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 ${B}^{0}\ensuremath{-}{B}^{0}$ oscillation frequency $\ensuremath{\Delta}{m}_{d}$ 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 $\ensuremath{\Delta}{m}_{d}=0.516\ifmmode\pm\else\textpm\fi{}0.016(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.010(\mathrm{syst}){\mathrm{ps}}^{\ensuremath{-}1}.$ The value of the asymmetry amplitude $\mathrm{sin}2\ensuremath{\beta}$ 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 $\mathrm{sin}2\ensuremath{\beta}=0.59\ifmmode\pm\else\textpm\fi{}0.14(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.05(\mathrm{syst}),$ demonstrating that CP violation exists in the neutral B meson system. We also determine the value of the CP violation parameter $|\ensuremath{\lambda}|=0.93\ifmmode\pm\else\textpm\fi{}0.09(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.03(\mathrm{syst}),$ which is consistent with the expectation of $|\ensuremath{\lambda}|=1$ for no direct CP violation.