A study of the two body decays B$^{+}\rightarrow X_{c\bar c}$K$^{+}$, where X$_{c\bar c}$ refers to one charmonium state, is reported by the BABAR experiment using a data sample of 424 fb$^{-1}$. The absolute determination of many branching fractions for these decays are significantly improved compared to previous BABAR measurements. Evidence is found at 3 $\sigma$ level for the decay B$^{+}\rightarrow X(3872)$K$^{+}$. The absolute branching fraction BR(B$^{+}\rightarrow X(3872)$K$^{+}$) = (2.1$\pm$0.6(stat)$\pm$0.3(syst))$\cdot$10$^{-4}$ is measured for the first time. From this, it follows that the branching fraction BR($X(3872)\
\rightarrow J/\psi\pi^+\pi^-$) is equal to (4.1$\pm$1.3)\%, supporting the hypothesis of a molecular component for this resonance.