We measure the mass of the B^0_s meson by reconstructing the decay
chain B^0_s -> J/psi phi, J/psi -> mu^+mu^-, and phi -> K^+K^-.  The data are
obtained from 19.3 pb^-1 of integrated luminosity of pbar p collisions at sqrt
s = 1.8 TeV using the Collider Detector at Fermilab.  A sample of 80,000
inclusive J/psi -> mu^+mu^- events is used to study systematic biases in track
reconstruction and to calibrate the momentum scale.  We reconstruct the
kinematically similar decays B^+ -> J/psi K^+ and B^0 -> J/psi K^*0 to study
the mass measurement technique used for the B^0_s meson.  Based on the
observation of 32 +/- 6 candidates, the mass of the B^0_s meson is measured to
be 5369.9 +/- 2.3 +/- 1.3 MeV/c^2 and the mass difference between B^0_s and
the average mass of B^+ and B^0 is determined to be 89.8 +/- 2.7 +/- 1.2
MeV/c^2.