  We present results on time-dependent CP-violating asymmetries in neutral B
decays to several CP eigenstates. The measurements use a data sample of about
88 million Y(4S) --> B Bbar 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 B0 or B0bar from its decay products. The amplitude of the CP-violating
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.
