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 BBbar 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/ψ, ψ(2S), and χc1, and the light meson in the decay is either a K, K*, or π0.
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