Measurement of the Branching Fractions for the Exclusive Decays of $B^0$ and $B^+ \to \bar{D}^{}D^{}K

We report the observation of 823+/-57 B^0 and 970+/-65 B^+ decays to doubly charmed final states Dbar^(*)D^(*)K, where Dbar^(*) and D^(*) are fully reconstructed and K is either a K+/- or a K^0. We use a sample of 82.3+/- 0.9 million BBbar 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 Dbar^(*)D^(*)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 Dbar^(*)D^(*)K$ are found to be BR(B^0 --> Dbar^(*) D^(*) K) = (4.3 +/- 0.3 stat +/- 0.6 syst)%, BR(B^+ --> Dbar^(*) D^(*) K) = (3.5 +/- 0.3 stat +/- 0.5 syst)%. A search for decays to orbitally excited $D_s$ states, B --> Dbar^(*) D_{sJ}^{+} (D_{sJ}^{+} --> D^(*)0K^+) is also performed.