This paper reports an updated measurement of the standard model CP violation parameter sin2β using the CDF Detector at Fermilab. The entire run I data sample of 110 pb-1 of proton-antiproton collisions at s√=1.8 TeV is used to identify a signal sample of ∼400 B⃗ J/ψK0S events, where J/ψ⃗ μ+μ- and K0S→π+π-. The flavor of the neutral B meson is identified at the time of production by combining information from three tagging algorithms: a same-side tag, a jet-charge tag, and a soft-lepton tag. A maximum likelihood fitting method is used to determine sin2β=0.79+0.41-0.44(stat+syst). This value of sin2β is consistent with the standard model prediction, based upon existing measurements, of a large positive CP-violating asymmetry in this decay mode.