We report on a search for flavor-changing neutral current decays
of B mesons B -> mu^+mu_6 K^\pm and B -> mu^+mu^- K^*0 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 -> mu^+mu^- K^\pm)
< 1.1 x 10^-5, and BR(B -> mu^+mu^- K^*0) < 2.1 x 10^-5 at 90% confidence
level, which are consistent with Standard Model expectations but leave little
room for non-standard physics.