We report measurements of B-meson decays into two- and three-body final states containing two charmed baryons using a sample of 230 million Y(4S) -> B Bbar decays. We find significant signals in two modes, measuring branching fractions BF(B^- -> \\Lambda_c^+ \\Lambda_c^- K^-) = (1.14 +- 0.15 +- 0.17 +- 0.60) x 10^{-3} and BF(B^- -> \\Xi_c^0 \\Lambda_c^-) x BF(\\Xi_c^0 -> \\Xi^- \\pi^+) = (2.08 +- 0.65 +- 0.29 +- 0.54) x 10^{-5}, where the uncertainties are statistical, systematic, and from the branching fraction BF(\\Lambda_c^+ -> p K^- \\pi^+), respectively. We also set upper limits at the 90% confidence level on two other modes: BF(B0bar -> \\Xi_c^+ \\Lambda_c^-) x BF(\\Xi_c^+ -> \\Xi^- \\pi^+ \\pi^+) \\Lambda_c^+ \\Lambda_c^- K0bar) \\Lambda_c^+ \\Lambda_c^- K^-.
