We report the first measurements of the doubly charmed baryonic B decays B --> Lambda c+ Lambda c- K. The B+ --> Lambda c+ Lambda c- K+ decay is observed with a branching fraction of (6.5(-0.9)(+1.0)+/-1.1+/-3.4)x10(-4) and a statistical significance of 15.4sigma. The B0 --> Lambda c+ Lambda c- K0 decay is observed with a branching fraction of (7.9(-2.3)(+2.9)+/-1.2+/-4.1)x10(-4) and a statistical significance of 6.6sigma. The branching fraction errors are statistical, systematic, and the error resulting from the uncertainty of the Lambda c+ --> pK- pi+ decay branching fraction. The analysis is based on 357 fb(-1) of data accumulated at the Upsilon(4S) resonance with the Belle detector at the KEKB asymmetric-energy e+ e- collider.