We report on a search for B0(s) --> mu+ mu- and B0(d) --> mu+
mu- decays in p anti-p collisions at s**(1/2) = 1.96 TeV using 171 pb-1 and 240
pb-1 of data collected respectively by the CDF and D0 experiments at the
Fermilab Tevatron Collider.  The decay rates of these rare processes are
sensitive to contributions from physics beyond the Standard Model.  The
results from the two experiments are a combined branching fraction limit of
B(B0(s) --> mu+ mu-) < 2.7 x 10-7 and a limit from the CDF experiment of
B(B0(d) --> mu+ mu-) < 1.5 x 10-7 at 90% confidence level.