We report branching fraction measurements for exclusive decays of charged and neutral B mesons into two-body final states containing a charmonium meson. We use a sample of 22.72 ± 0:36 million BB events collected between October 1999 and October 2000 with the BABAR detector at the PEP-II storage rings at the Stanford Linear Accelerator Center. The charmonium mesons considered here are J/y , y (2S), and c_c1, and the light meson in the decay is either a K, K*, or p⁰.