We report the first observation of the $Xi_{c}(2930)^0$ charmed-strange baryon with a significance greater than 5$sigma$. The $Xi_{c}(2930)^0$ is found in its decay to $K^- Lambda_{c}^+$ in $B^{-} to K^{-} Lambda_{c}^{+} bar{Lambda}_{c}^{-}$ decays. The measured mass and width are $[2928.9 pm 3.0(rm stat.)^{+0.9}_{-12.0}(rm syst.)]$ MeV/$c^{2}$ and $[19.5 pm 8.4(rm stat.) ^{+5.9}_{-7.9}(rm syst.)]$ MeV, respectively, and the product branching fraction is ${cal B}(B^{-} to Xi_{c}(2930)^0 bar{Lambda}_{c}^{-}) {cal B}(Xi_{c}(2930)^0 to K^- Lambda_{c}^{+})=[1.73 pm 0.45(rm stat.) pm 0.21(rm syst.)]times 10^{-4}$. We also measure ${cal B}(B^{-} to K^{-} Lambda_{c}^{+} bar{Lambda}_{c}^{-}) = [4.80 pm 0.43(rm stat.) pm 0.60(rm syst.)] times 10^{-4}$ with improved precision, and search for the charmonium-like state $Y(4660)$ and its spin partner, $Y_{eta}$, in the $Lambda_{c}^{+}bar{Lambda}_{c}^{-}$ invariant mass spectrum. No clear signals of the $Y(4660)$ nor its spin partner are observed and the 90% credibility level (C.L.) upper limits on their production rates are determined. These measurements are obtained from a sample of $(772pm11)times 10^{6} Bbar{B}$ pairs collected at the $Upsilon(4S)$ resonance by the Belle detector at the KEKB asymmetric energy electron-positron collider.