We measure the mass of the B-s(0) meson by reconstructing the decay chain B-s(0)-->J/psi phi, J/psi-->mu(+)mu(-), and phi-->K+K-. The data are obtained from 19.3 pb(-1) of integrated luminosity of <(p) over bar p> collisions at root 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/mu K+ and B-0-->J/psi K*(0) to study the mass measurement technique used for the B-s(0) meson. Based on the observation of 32 +/- 6 candidates, the mass of the B-s(0) meson is measured to be 5369.9 +/- 2.3 +/- 1.3 MeV/c(2) and the mass difference between B-s(0) and the average mass of B+ and B-0 is determined to be 89.7 +/- 2.7 +/- 1.2 MeV/c(2). RI Vucinic, Dejan/C-2406-2008; Chiarelli, Giorgio/E-8953-2012; Azzi, Patrizia/H-5404-2012