We measure the mass of the B^0_s meson by fully reconstructing
the decay B^0_s -> J/psi phi with the subsequent decays J/psi -> mu^+ mu^-
and phi -> K^+ K^-.  The data was 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 (CDF).  A sample of 80,000 inclusive J/psi -> mu^+ mu^-
events is used to study systematic biases in track reconstruction and is
used to calibrate the momentum scale.  We reconstruct the kinematically
similar decays B^- -> J/psi K^- and B^0 -> J/psi K^* to study the mass
measurement technique used for the B^0_s meson.  Based on the observation
of 32 \pm 6 candidates, the mass of the B^0_s meson is measured to be
5369.9 \pm 2.3 \pm 1.3 MeV/c^2.