We report on the reconstruction of the decay mode B0→J/ψKS0 using 19.3pb−1 of data collected by the Collider Detector at Fermilab in p¯p collisions at s=1.8TeV. 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/ψK+, B0→J/ψK*(892)0, and B+→J/ψK*(892)+ are reconstructed. We measure three ratios of branching ratios, each one relative to the B+→J/ψK+ mode. We also report the ratio of decay rates, γ(B→J/ψK*)/γ(B→J/ψK), for the vector-vector relative to the vector-pseudoscalar modes, to be 1.32±0.23(stat)±0.16(syst).