  This paper reports an updated measurement of the Standard Model CP violation
parameter $\sin2\beta$ using the CDF Detector at Fermilab. The entire Run I
data sample of 110 pb$^{-1}$ of proton antiproton collisions at $\sqrt{s}=1.8
TeV$ is used to identify a signal sample of $\sim 400$ $B \to J/\psi K^0_S$
events, where $J/\psi \to \mu^+\mu^-$ and $K_S^0 \to \pi^+\pi^-$. 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
$\sin2\beta = 0.79 {+0.41\atop-0.44}$(stat+syst). This value of $\sin 2 \beta$
is consistent with the Standard Model prediction, based upon existing
measurements, of a large positive CP violating asymmetry in this decay mode.
