Measurement of the branching fractions for the exclusive decays of $B^0$ and $B^{+}$ to $D^{\left(*\right)}{D}^{\left(*\right)}K$

We report the observation of $823\pm 57$ $B^0$ and $970\pm 65$ $B^{+}$ decays to doubly charmed final states $D^{\left(*\right)}{D}^{\left(*\right)}K,$ where $D^{\left(*\right)}$ and $D^{\left(*\right)}$ are fully reconstructed and K is either a $K^{\pm }$ or a $K^0.$ We use a sample of $82.3\pm 0.9$ million $B\overline{B}$ events collected between 1999 and 2002 with the BABAR detector at the PEP-II storage ring at the Stanford Linear Accelerator Center. The 22 possible B decays to $D^{\left(*\right)}{D}^{\left(*\right)}K$ are reconstructed exclusively and the corresponding branching fractions or limits are determined. The branching fractions of the $B^0$ and of the $B^{+}$ to $D^{\left(*\right)}{D}^{\left(*\right)}K$ are found to be $\mathcal{B}{(B}^0\to D^{\left(*\right)}{D}^{\left(*\right)}K)=[4.3\pm 0.3\left(\mathrm{stat}\right)\pm 0.6\left(\mathrm{syst}\right)]\%,$ $\mathcal{B}{(B}^{+}\to D^{\left(*\right)}{D}^{\left(*\right)}K)=[3.5\pm 0.3\left(\mathrm{stat}\right)\pm 0.5\left(\mathrm{syst}\right)]\%.\mathrm{}$ A search for decays to orbitally excited $D_s$ states, $\overrightarrow{B}{D}^{\left(*\right)}{D}_{\mathrm{sJ}}^{+}$ ${(D}_{\mathrm{sJ}}^{+}\to D^{\left(*\right)0\mathrm{}}{K}^{+}),$ is also performed. No statistically significant contributions from $D_{s1\mathrm{}}^{+}\mathrm{}\left(2536\right)\mathrm{}\to D^{*0}{K}^{+}$ and $D_{\mathrm{sJ}}^{+}\mathrm{}\left(2573\right)\mathrm{}\to D^0{K}^{+}$ to the $D^{\left(*\right)}{D}^{\left(*\right)0\mathrm{}}{K}^{+}$ final state are found and we set 90% C.L. limits on their production rates.