Two structures are observed close to the kinematic threshold in the Ξ(b)(0)π(-) mass spectrum in a sample of proton-proton collision data, corresponding to an integrated luminosity of 3.0 fb(-1), recorded by the LHCb 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 Ξ(b)('-) and Ξ(b)(*-). Interpreting the structures as these resonances, we measure the mass differences and the width of the heavier state to be m(Ξ(b)('-))-m(Ξ(b)(0))-m(π(-))=3.653±0.018±0.006 MeV/c(2), m(Ξ(b)(*-))-m(Ξ(b)(0))-m(π(-))=23.96±0.12±0.06 MeV/c(2), Γ(Ξ(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 Γ(Ξ(b)('-))<0.08 MeV at 95% confidence level. Relative production rates of these states are also reported.