  Recently Belle has established the 90% confidence level (CL) upper limit $\b
< 9.4 \times 10^{-7}$ for the branching ratio for $B^0\to J/\psi \phi$, a
process expected to be suppressed by the Okubo-Zweig-Iizuka (OZI) rule
disfavoring disconnected quark diagrams. We use information on $\omega$--$\phi$
mixing to establish likely lower bounds on this and related processes. We find
that the Belle result is about a factor of five above our limit, while other
decays such as $B^0 \to \bar D^0 \phi$ and $B^+ \to \pi^+ \phi$, for which
upper limits have been obtained by BaBar, could be observable with similar
improvements in data. We argue that a significant enhancement of our predicted
decay rates by rescattering is unlikely.
