We present a measurement of the branching fractions of the 22 decay channels of the B^0 and B+ mesons to D̅ ^((*))D^((*))K, where the D^((*)) and D̅ ^((*)) mesons are fully reconstructed. Summing the 10 neutral modes and the 12 charged modes, the branching fractions are found to be B(B^0→D̅6((*))D^((*))K)=(3.68 ± 0.10 ± 0.24)% and B(B^+→D̅ ^((*))D^((*))K)=(4.05 ± 0.11 ± 0.28)%, where the first uncertainties are statistical and the second systematic. The results are based on 429 fb^(-1) of data containing 471 × 10^6BB̅ pairs collected at the Υ(4S) resonance with the BABAR detector at the SLAC National Accelerator Laboratory
