We report on a search for flavor-changing neutral current decays of B
mesons into $\mu \mu K^{\pm}$ and $\mu \mu K^{*}$ using data obtained in
the Collider Detector at Fermilab (CDF) 1992-1993 data taking run. To
reduce the amount of background in our data we use precise tracking
information from the CDF silicon vertex detector to pinpoint the location
of the decay vertex of the B candidate, and accept only events which have
a large decay time.   We compare this data to a B meson signal obtained
in a similar fashion, but where the muon pairs originate from $\psi$ decays, 
and calculate the relative branching ratios. In absence of any indication 
of flavor-changing neutral current decays we set an upper limits of
BR($B \rightarrow \mu \mu K^{\pm}$) $< 3.5 \times 10^{-5}$, and
BR($B \rightarrow \mu \mu K^{*}$) $< 5.1 \times 10^{-5}$ at 90\%
confidence level, which are consistent with Standard Model expectations
but leave little room for non-standard physics.