We present results on time-dependent CP asymmetries in neutral B decays to several CP eigenstates. The measurements use a data sample of about 227 million Upsilon(4S) -> BBbar decays collected by the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC. The amplitude of the CP asymmetry, sin2Beta in the Standard Model, is derived from decay-time distributions from events in which one neutral B meson is fully reconstructed in a final state containing a charmonium meson and the other $B$ meson is determined to be either a B0 or B0bar from its decay products. We measure sin2Beta = 0.722 +/- 0.040 (stat.) +/- 0.23 (syst.) in agreement with the Standard Model expectation.