  We report on measurements of the decays of $\overline{B}$ mesons into the
semileptonic final states $\overline{B} \to D^{(*)} \pi^+\pi^-
\ell^-\overline{\nu}$, where $D^{(*)}$ represents a $D$ or $D^*$ meson and
$\ell^-$ is an electron or a muon. These measurements are based on $471\times
10^6$ $B\overline{B}$ pairs recorded with the $BABAR$ detector at the SLAC
asymmetric $B$ Factory PEP-II. We determine the branching fraction ratios
$R^{(*)}_{\pi^+\pi^-} = \mathcal{B}(\overline{B} \to D^{(*)} \pi^+\pi^- \ell^-
\overline{\nu})/\mathcal{B}(\overline{B} \to D^{(*)} \ell^-\overline{\nu})$
using events in which the second $B$ meson is fully reconstructed. We find
$R_{\pi^+\pi^-} = 0.067 \pm 0.010 \pm 0.008$ and $R^*_{\pi^+\pi^-} = 0.019 \pm
0.005 \pm 0.004$, where the first uncertainty is statistical and the second is
systematic. Based on these results, we estimate that $\overline{B} \to
D^{(*)}\pi\pi\ell^-\overline{\nu}$ decays, where $\pi$ denotes both a $\pi^\pm$
and $\pi^0$ meson, account for up to half the difference between the measured
inclusive semileptonic branching fraction to charm hadrons and the
corresponding sum of previously measured exclusive branching fractions.
