  The CKM mechanism is well established as the dominant mechanism for $CP$
violation, which was first discovered in the neutral kaons in 1964
\cite{ref:cpkaon}. To search for new sources of CP violation, one can exploit a
handful of systems in which the standard model makes a precise prediction of CP
violation. In the $B^0_s\to J/\psi \phi$ system, CP violation in the
interference of mixing and decay is precisely predicted in the standard model,
the prediction being very close to zero. The CDF experiment reconstructs about
2000 signal events in 1.35 fb$^{-1}$ of luminosity. We obtain a confidence
region in the space of the parameters $\beta_s$, the CP phase, and
$\Delta\Gamma$, the width difference. This result is 1.5 $\sigma$ from the
standard model prediction.
