A study of the two-body decays B^{±}→X_{cc[over ¯]}K^{±}, where X_{cc[over ¯]} refers to one charmonium state, is reported by the BABAR Collaboration using a data sample of 424 fb^{-1}. The absolute determination of branching fractions for these decays are significantly improved compared to previous BABAR measurements. Evidence is found for the decay B^{+}→X(3872)K^{+} at the 3σ level. The absolute branching fraction B[B^{+}→X(3872)K^{+}]=[2.1±0.6(stat)±0.3(syst)]×10^{-4} is measured for the first time. It follows that B[X(3872)→J/ψπ^{+}π^{-}]=(4.1±1.3)%, supporting the hypothesis of a molecular component for this resonance.