We report a measurement of the ratios of the decay rates of the B~+, B~0 and B~0_s mesons into exclusive final states containing a J/psi meson. The final states were selected from 19.6 pb~{-1} of p-pbar collisions recorded by the Collider Detector at Fermilab. These data are interpreted to determine the bquark fragmentation fractions f_u, f_d and f_s. We also determine the branching fractions for the decay modes B~+ --> J/psi K~+, B~+ --> J/psi K~*(892)~+, B~0 --> J/psi K~0, B~0 --> J/psi K~*(892)~0 and B_s~0 --> J/psi phi(1020). We discuss the implications of these measurements to B meson decay models.