  We present the first measurements of branching fractions of rare tau-lepton
decays, $\tau^- \rightarrow \pi^- \nu_{\tau} \ell^+ \ell^-$ ($\ell = e$ 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 $\tau^- \rightarrow \pi^- \nu_\tau
e^+ e^-$ 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 $\mathcal{B}(\tau^-\rightarrow
\pi^- \nu_\tau e^+ e^-)[M_{\pi^- e^+ e^-}>1.05~{\rm GeV}/c^2] = (5.90 \pm 0.53
\pm 0.85 \pm 0.11) \times 10^{-6}$, 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 $\mathcal{B}_{A}(\tau^-\rightarrow \pi^-
\nu_\tau e^+ e^-) = (1.46 \pm 0.13 \pm 0.21) \times 10^{-5}$ to
$\mathcal{B}_{V}(\tau^-\rightarrow \pi^- \nu_\tau e^+ e^-) = (3.01 \pm 0.27 \pm
0.43) \times 10^{-5}$, respectively. An upper limit is set on the branching
fraction of the $\tau^- \rightarrow \pi^- \nu_\tau \mu^+ \mu^-$ decay,
$\mathcal{B}(\tau^-\rightarrow \pi^- \nu_\tau \mu^+ \mu^-) < 1.14 \times
10^{-5}$, at the 90% confidence level.
