We report the observation of 823±57 B0 and 970±65 B+ decays to doubly charmed final states D(*)D(*)K, where D(*) and D(*) are fully reconstructed and K is either a K± or a K0. We use a sample of 82.3±0.9 million BB̅ 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(*)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 D(*)D(*)K are found to be B(B0→D(*)D(*)K)=[4.3±0.3(stat)±0.6(syst)]%, B(B+→D(*)D(*)K)=[3.5±0.3(stat)±0.5(syst)]%. A search for decays to orbitally excited Ds states, B⃗D(*)DsJ+ (DsJ+→D(*)0K+), is also performed. No statistically significant contributions from Ds1+(2536)→D*0K+ and DsJ+(2573)→D0K+ to the D(*)D(*)0K+ final state are found and we set 90% C.L. limits on their production rates.
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