The first observation of the semileptonic b-baryon decay Lambda(0)(b) -> Lambda(+)(c) tau(-)(nu) over bar (tau) with a significance of 6.1 sigma, 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 tau(-) lepton is reconstructed in the hadronic decay to three charged pions. The ratio K = B(Lambda(0)(b) -> Lambda(+)(c) tau(-)(nu) over bar (tau))/B(Lambda(0)(b) -> Lambda(+)(c)pi(-)pi(+)pi(-)) is measured to be 2.46 +/- 0.27 +/- 0.40, where the first uncertainty is statistical and the second systematic. The branching fraction B(Lambda(0)(b) -> Lambda(+)(c) tau(-)(nu) over bar (tau)) (1.50 +/- 0.16 +/- 0.25 +/- 0.23)% is obtained, where the third uncertainty is from the external branching fraction of the normalization channel Lambda(0)(b) -> Lambda(+)(c)pi(-)pi(+)pi(-). The ratio of semileptonic branching fractions R(Lambda(+)(c)) B(Lambda(0)(b) -> Lambda(+)(c) tau(-)(nu) over bar (tau))/B(Lambda(0)(b) -> Lambda(+)(c)mu(-)(nu) over bar (tau)) is derived to be 0.242 +/- 0.026 +/- 0.040 +/- 0.059, where the external branching fraction uncertainty from the channel Lambda(0)(b) -> Lambda(+)(c)mu(-)(nu) over bar (tau) contributes to the last term. This result is in agreement with the standard model prediction.