A study of the two-body decays B±→XccK±, where Xcc 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.