We present measurements of time-dependent CP-violating asymmetries in neutral B decays to several CP eigenstates. The measurement uses a data sample of 23×106 ϒ\(4S\)-->BB̄ decays collected by the BABAR detector at the PEP-II asymmetric B Factory at SLAC. In this sample, we find events in which one neutral B meson is fully reconstructed in a CP eigenstate containing charmonium and the flavor of the other neutral B meson is determined from its decay products. The amplitude of the CP-violating asymmetry, which in the standard model is proportional to sin2β, is derived from the decay time distributions in such events. The result is sin2β = 0.34+/-0.20 \(stat\)+/-0.05 \(syst\).