Observation ofCPViolation in theB0Meson System

We present an updated measurement of time-dependent $\mathrm{CP}$-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 $\Upsilon \left(4S\right)$ decays collected in 2001, bringing the data available to $32\times {10}^6$ $B\overline{B}$ 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 $\mathrm{CP}$-violating asymmetry, which in the standard model is proportional to $\sin 2\beta$, is derived from the decay time distributions in such events. The result $\sin 2\beta =0.59\mathrm{}\pm 0.14\left(\mathrm{stat}\right)\pm 0.05\left(\mathrm{syst}\right)$ establishes $\mathrm{CP}$ violation in the $B^0$ meson system. We also determine $|\lambda |=0.93\mathrm{}\pm 0.09\left(\mathrm{stat}\right)\pm 0.03\left(\mathrm{syst}\right)$, consistent with no direct $\mathrm{CP}$ violation.