  We present an analysis of the exclusive
$B^{+}\to\pi^{+}\pi^{-}\ell^{+}\nu_{\ell}$ decay, where $\ell$ represents an
electron or a muon, with the assumption of charge-conjugation symmetry and
lepton universality. The analysis uses the full $\Upsilon(4S)$ data sample
collected by the Belle detector, corresponding to 711 fb$^{-1}$ of integrated
luminosity. We select the events by fully reconstructing one $B$ meson in
hadronic decay modes, subsequently determining the properties of the other $B$
meson. We extract the signal yields using a binned maximum-likelihood fit to
the missing-mass squared distribution in bins of the invariant mass of the two
pions or the momentum transfer squared. We measure a total branching fraction
of ${{\cal B}(B^{+}\to \pi^{+}\pi^{-}\ell^{+}\nu_{\ell})= [22.7 ^{+1.9}_{-1.6}
(\mathrm{stat}) \pm 3.5(\mathrm{syst}) ]\times 10^{-5}}$, where the
uncertainties are statistical and systematic, respectively. This result is the
first reported measurement of this decay.
