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^+) < 5.6 x 10^{-5} and BF(B0bar -> \Lambda_c^+ \Lambda_c^- K0bar) < 1.5 x 10^{-3}. We observe structure centered at an invariant mass of 2.93 GeV/c^2 in the \Lambda_c^+ K^- mass distribution of the decay B^- -> \Lambda_c^+ \Lambda_c^- K^-.

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