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 x 10(6) upsilon(4S) --> BB decays collected by the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC. The amplitude of the CPasymmetry, 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 B0 from its decay products. We measure sin2beta = 0.722 +/- 0.040(stat) +/- 0.023(syst) in agreement with the standard model expectation.