The first observation of the semileptonic b-baryon decay Λb0→Λc+τ-ν¯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(Λb0→Λc+τ-ν¯τ)/B(Λb0→Λc+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(Λb0→Λc+τ-ν¯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 Λb0→Λc+π-π+π-. The ratio of semileptonic branching fractions R(Λc+)B(Λb0→Λc+τ-ν¯τ)/B(Λb0→Λc+μ-ν¯μ) is derived to be 0.242±0.026±0.040±0.059, where the external branching fraction uncertainty from the channel Λb0→Λc+μ-ν¯μ contributes to the last term. This result is in agreement with the standard model prediction.