  We have measured the time-dependent decay rate for the process $B \to J/\psi
K^{*0}(892)$ in a sample of about 88 million $\Upsilon(4S) \to B\bar{B}$ decays
collected with the BaBar detector at the PEP-II asymmetric-energy $B$ Factory
at SLAC. In this sample we study flavor-tagged events in which one neutral $B$
meson is reconstructed in the $J/\psi K^{*0}$ or $J/\psi \bar{K}^{*0}$ final
state. We measure the coefficients of the cosine and sine terms in the
time-dependent asymmetries for $J/\psi K^{*0}$ and $J/\psi \bar{K}^{*0}$, find
them to be consistent with the Standard Model expectations, and set upper
limits at 90% C.L. on the decay amplitude ratios $|A (\bar{B}^0 \to J/\psi
K^{*0})|/|A (B^0 \to J/\psi K^{*0})| < 0.26$ and $|A (B^0 \to J/\psi
\bar{K}^{*0})|/|A (\bar{B}^0 \to J/\psi \bar{K}^{*0})| <0.32$. For a single
ratio of wrong-flavor to favored amplitudes for $B^0$ and $\bar{B}^0$ combined,
we obtain an upper limit of 0.25 at 90% C.L.
