Using a pp collision data sample corresponding to an integrated luminosity of 3.0 fb^{-1}, collected by the LHCb detector, we present the first search for the strangeness-changing weak decay Ξ_{b}^{-}→Λ_{b}^{0}π^{-}. No b hadron decay of this type has been seen before. A signal for this decay, corresponding to a significance of 3.2 standard deviations, is reported. The relative rate is measured to be f_{Ξ_{b}^{-}}/f_{Λ_{b}^{0}}B(Ξ_{b}^{-}→Λ_{b}^{0}π^{-})=(5.7±1.8_{-0.9}^{+0.8})×10^{-4},where f_{Ξ_{b}^{-}} and f_{Λ_{b}^{0}} are the b→Ξ_{b}^{-} and b→Λ_{b}^{0} fragmentation fractions, and B(Ξ_{b}^{-}→Λ_{b}^{0}π^{-}) is the branching fraction. Assuming f_{Ξ_{b}^{-}}/f_{Λ_{b}^{0}} is bounded between 0.1 and 0.3, the branching fraction B(Ξ_{b}^{-}→Λ_{b}^{0}π^{-}) would lie in the range from (0.57±0.21)% to (0.19±0.07)%.