Two structures are observed close to the kinematic threshold in the Xi(0)(b)pi(-) mass spectrum in a sample of proton-proton collision data, corresponding to an integrated luminosity of 3.0 fb(-1), recorded by the LEICb experiment. In the quark model, two baryonic resonances with quark content bds are expected in this mass region: the spin-parity J(P) = (1/2)(+) and J(P) = (3/2)(+) states, denoted Xi'(-)(b) and Xi*(-)(b). Interpreting the structures as these resonances, we measure the mass differences and the width of the heavier state to be m(Xi'(-)(b)) - m(Xi(0)(b)) - m(pi(-))=3.653 +/- 0.018 +/- 0.006 MeV/c(2), m(Xi*(-)(b)) - m(Xi(0)(b)) - m(pi(-)) = 23.96 +/- 0.12 +/- 0.06 MeV/e(2), Gamma(Xi*(-)(b)) = 1.65 +/- 0.31 +/- 0.10 MeV, where the first and second uncertainties are statistical and systematic, respectively. The width of the lighter state is consistent with zero, and we place an upper limit of Gamma(Xi'(-)(b)) <0.08 MeV at 95% confidence level. Relative production rates of these states are also reported.