We present the first measurements of branching fractions of rare tau-lepton decays, $tau^- rightarrow pi^- u_{tau} ell^+ ell^-$ ($ell = e$ or $mu$), using a data sample corresponding to 562 fb$^{-1}$ collected at a center-of-mass energy of 10.58 GeV with the Belle detector at the KEKB asymmetric-energy $e^+ e^-$ collider. The $tau^- rightarrow pi^- u_tau e^+ e^-$ decay is observed for the first time with 7.0$sigma$ significance. The partial branching fraction determined by the structure-dependent mechanisms mediated by either a vector or an axial-vector current for the mass region $M_{pi e e}>1.05$ GeV/$c^2$ is measured to be $mathcal{B}(tau^-rightarrow pi^- u_tau e^+ e^-)[M_{pi^- e^+ e^-}>1.05~{rm GeV}/c^2] = (5.90 pm 0.53 pm 0.85 pm 0.11) times 10^{-6}$, where the first uncertainty is statistical, the second is systematic, and the third is due to the model dependence. In the full phase space, due to the different detection efficiencies for the structure-dependent mechanisms mediated by axial-vector and vector currents, the branching fraction varies from $mathcal{B}_{A}(tau^-rightarrow pi^- u_tau e^+ e^-) = (1.46 pm 0.13 pm 0.21) times 10^{-5}$ to $mathcal{B}_{V}(tau^-rightarrow pi^- u_tau e^+ e^-) = (3.01 pm 0.27 pm 0.43) times 10^{-5}$, respectively. An upper limit is set on the branching fraction of the $tau^- rightarrow pi^- u_tau mu^+ mu^-$ decay, $mathcal{B}(tau^-rightarrow pi^- u_tau mu^+ mu^-) < 1.14 times 10^{-5}$, at the 90% confidence level.
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