  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 B Factory at the Stanford Linear Accelerator Center. The data
sample consists of 29.7 ${\rm fb}^{-1}$ recorded at the $\Upsilon(4S)$
resonance and 3.9 ${\rm fb}^{-1}$ off-resonance. One of the neutral B mesons,
which are produced in pairs at the $\Upsilon(4S)$, is fully reconstructed in
the CP decay modes $J/\psi K^0_S$, $\psi(2S) K^0_S$, $\chi_{c1} K^0_S$, $J/\psi
K^{*0}$ ($K^{*0}\to K^0_S\pi^0$) and $J/\psi K^0_L$, or in flavor-eigenstate
modes involving $D^{(*)}\pi/\rho/a_1$ and $J/\psi K^{*0}$ ($K^{*0}\to
K^+\pi^-$). 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. The proper
time elapsed between the decays is determined by measuring the distance between
the decay vertices. A maximum-likelihood fit to this flavor eigenstate sample
finds $\Delta m_d = 0.516\pm 0.016 {\rm (stat)} \pm 0.010 {\rm (syst)} {\rm
ps}^{-1}$. The value of the asymmetry amplitude $\sin2\beta$ 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 $\sin2\beta=0.59\pm 0.14 {\rm (stat)} \pm 0.05
{\rm (syst)}$, demonstrating that CP violation exists in the neutral B meson
system. (abridged)
