We present results on time-dependent CP asymmetries in neutral B decays to several CP eigenstates. The measurements use a data sample of about 88 * 106 U(4S) -> B.hivin.B decays collected between 1999 and 2002 with the BABAR detector at the PEP-II asym.-energy B factory at SLAC. We study events in which one neutral B meson is fully reconstructed in a final state contg. a charmonium meson and the other B meson is detd. to be either a B0 or .hivin.B0 from its decay products. The amplitude of the CP asymmetry, which in the std. model is proportional to sin2b, is derived from the decay-time distributions in such events. We measure sin2b = 0.741 +- 0.067(stat) +- 0.034(syst) and |l| = 0.948 +- 0.051(stat) +- 0.030(syst). The magnitude of l is consistent with unity, in agreement with the std. model expectation of no direct CP violation in these modes. [on SciFinder (R)]