We report the observation of 823±57 B0 and 970±65 B+ decays to doubly charmed final states D̄ (*)D(*)K, where D̄(*) and D(*) are fully reconstructed and K is either a K ± or a K0. We use a sample of 82.3 ±0.9 million B B̄ events collected between 1999 and 2002 with the BABAR detector at the PEP-II storage ring at the Stanford Linear Accelerator Center. The 22 ossible B decays to D̄(*)D(*)K are reconstructed exclusively and the corresponding branching fractions or limits are determined. The branching fractions of the B0 and of the B + to D̄(*)D(*)K are found to be Β(B0→D̄(*)D(*)K) = [4.3±0.3(stat)±0.6(syst)]%, Β(B+→D̄ (*)D(*)K) = [3.5±0.3(stat)±0. 5(syst)]%.A search for decays to orbitally excited Ds states, B→D̄(*)DsJ+(DsJ+→D(*)0K+), is also performed. No statistically significant-contributions from Ds1+(2536) →D*0K+ and DsJ+(2573) →D0K+ to the D̄(*)D (*)0K+ final state are found and we set 90% C.L. limits on their production rates. © 2003 The American Physical Society.