We report on the reconstruction of the decay mode B-0 --> J/psi K-S(0) using 19.3 pb(-1) of data collected by the Collider Detector at Fermilab in <(p)over bar p> collisions at root s = 1.8 TeV. A signal of 41.8 +/- 6.9 events, with a signal-to-background ratio of 9:1, is observed. Three additional decay modes B+ --> J/psi K+, B-0 --> J/psi K*(892)(0), and B+ --> J/psi K*(892)(+) are reconstructed. We measure three ratios of branching ratios, each one relative to the B+ --> J/psi K+ mode. We also report the ratio of decay rates, Gamma(B --> J/psi K*)/Gamma(B --> J/psi K), for the vector-vector relative to the vector-pseudoscalar modes, to be 1.32 +/- 0.23(stat) +/- 0.16(syst). RI Chiarelli, Giorgio/E-8953-2012; Azzi, Patrizia/H-5404-2012