We present a measurement of the branching fractions of the 22 decay channels B0 and B+ to Dbar^(*) D^(*) K, where Dbar^(*) and D^(*) are fully reconstructed. The B0 and B+ mesons are reconstructed in a sample of hadronic events for all the possible Dbar D K modes, namely B0 -> D^(*)- D^(*)0 K^+, D^(*)-D^(*)+ K^0, Dbar^(*)0 D^(*)0 K0 and B^+ -> Dbar^(*)0 D^(*)+ K0, Dbar^(*)0 D^(*)0 K+, D^(*)- D^(*)+ K+. The results are based on 423 fb-1 of data that contained 465 10^6 BBbar pairs collected at the Upsilon(4S) resonance with the BaBar detector at the PEP-II B factory.