The first observation of the semileptonic b-baryon decay Λ_{b}^{0}→Λ_{c}^{+}τ^{-}ν[over ¯]_{τ}, with a significance of 6.1σ, is reported using a data sample corresponding to 3 fb^{-1} of integrated luminosity, collected by the LHCb experiment at center-of-mass energies of 7 and 8 TeV at the LHC. The τ^{-} lepton is reconstructed in the hadronic decay to three charged pions. The ratio K=B(Λ_{b}^{0}→Λ_{c}^{+}τ^{-}ν[over ¯]_{τ})/B(Λ_{b}^{0}→Λ_{c}^{+}π^{-}π^{+}π^{-}) is measured to be 2.46±0.27±0.40, where the first uncertainty is statistical and the second systematic. The branching fraction B(Λ_{b}^{0}→Λ_{c}^{+}τ^{-}ν[over ¯]_{τ})=(1.50±0.16±0.25±0.23)% is obtained, where the third uncertainty is from the external branching fraction of the normalization channel Λ_{b}^{0}→Λ_{c}^{+}π^{-}π^{+}π^{-}. The ratio of semileptonic branching fractions R(Λ_{c}^{+})≡B(Λ_{b}^{0}→Λ_{c}^{+}τ^{-}ν[over ¯]_{τ})/B(Λ_{b}^{0}→Λ_{c}^{+}μ^{-}ν[over ¯]_{μ}) is derived to be 0.242±0.026±0.040±0.059, where the external branching fraction uncertainty from the channel Λ_{b}^{0}→Λ_{c}^{+}μ^{-}ν[over ¯]_{μ} contributes to the last term. This result is in agreement with the standard model prediction.