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 x 10(6) Upsilon(4S)-->B(-)B decays collected between 1999 and 2002 with the BABAR detector at the PEP-II asymmetric-energy B factory at SLAC. We study 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 B(0) or B(-0) from its decay products. The amplitude of the CP asymmetry, which in the standard model is proportional to sin2beta, is derived from the decay-time distributions in such events. We measure sin2beta=0.741+/-0.067(stat)+/-0.034(syst) and |lambda|=0.948+/-0.051(stat)+/-0.030(syst). The magnitude of lambda is consistent with unity, in agreement with the standard model expectation of no direct CP violation in these modes.