  The first observation of the semileptonic $b$-baryon decay $ \Lambda_b^0
\rightarrow \Lambda_c^+ \tau^-\overline{\nu}_{\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 centre-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 branching fraction
${\mathcal{B}}(\Lambda_b^0 \rightarrow \Lambda_c^+\tau^-\overline{\nu}_{\tau})
= (1.50 \pm 0.16\pm 0.25\pm 0.23)\%$ is obtained, where uncertainties are
statistical, systematic and from the external branching fraction of the
normalisation channel $\Lambda_b^0\rightarrow \Lambda_c^+\pi^-\pi^+\pi^-$. The
ratio of semileptonic branching fractions ${\mathcal{R}}( \Lambda_c^+)\equiv
{\mathcal{B}}( \Lambda_b^0 \rightarrow \Lambda_c^+
\tau^-\overline{\nu}_{\tau})/{\mathcal{B}}( \Lambda_b^0 \rightarrow \Lambda_c^+
\mu^-\overline{\nu}_{\mu})$ is derived to be $0.242 \pm 0.026 \pm 0.040\pm
0.059$, where the external branching fraction uncertainty from the channel
$\Lambda_b^0\rightarrow \Lambda_c^+\mu^-\overline{\nu}_{\mu}$ contributes to
the last term. This result is in agreement with the Standard Model prediction.
