  An angular analysis of the decay ${\overline{B} \rightarrow D^\ast \ell^-
\overline{\nu}_\ell}$, $\ell\in\{e,\mu\}$, is reported using the full $e^+e^-$
collision data set collected by the \babar experiment at the $\Upsilon(4S)$
resonance. One $B$ meson from the ${\Upsilon(4S)\to B\overline{B}}$ decay is
fully reconstructed in a hadronic decay mode, which constrains the kinematics
and provides a determination of the neutrino momentum vector. The kinematics of
the semileptonic decay is described by the di-lepton mass squared, $q^2$, and
three angles. The first unbinned fit to the full four-dimensional decay rate in
the Standard Model is performed in the so-called Boyd-Grinstein-Lebed approach,
which employs a generic $q^2$ parameterization of the underlying form factors
based on crossing symmetry, analyticity and QCD dispersion relations for the
amplitudes. A fit using the more model-dependent Caprini-Lellouch-Neubert (CLN)
approach is performed as well. Our form factor shapes show deviations from
previous fits based on the CLN parameterization. The latest form factors also
provide an updated prediction for the branching fraction ratio
$\mathcal{R}(D^\ast)\equiv \mathcal{B}(\overline{B}\to D^\ast \tau^-
\bar{\nu}_\tau)/\mathcal{B}({\overline{B} \rightarrow D^\ast \ell^-
\overline{\nu}_\ell})=0.253 \pm 0.005$. Finally, using the well measured
branching fraction for the ${\overline{B} \rightarrow D^\ast \ell^-
\overline{\nu}_\ell}$ decay, a value of $|V_{cb}|=(38.36\pm 0.90)\times10^{-3}$
is obtained that is consistent with the current world average for exclusive
${\overline{B}\to D^{(\ast)}\ell^- \overline{\nu}_\ell}$ decays and remains in
tension with the determination from inclusive semileptonic $B$ decays to final
states with charm.
