  We present the first unquenched lattice-QCD calculation of the hadronic form
factors for the exclusive decay $\overline{B} \rightarrow D \ell
\overline{\nu}$ at nonzero recoil. We carry out numerical simulations on
fourteen ensembles of gauge-field configurations generated with 2+1 flavors of
asqtad-improved staggered sea quarks. The ensembles encompass a wide range of
lattice spacings (approximately 0.045 to 0.12 fm) and ratios of light (up and
down) to strange sea-quark masses ranging from 0.05 to 0.4. For the $b$ and $c$
valence quarks we use improved Wilson fermions with the Fermilab
interpretation, while for the light valence quarks we use asqtad-improved
staggered fermions. We extrapolate our results to the physical point using
rooted staggered heavy-light meson chiral perturbation theory. We then
parameterize the form factors and extend them to the full kinematic range using
model-independent functions based on analyticity and unitarity. We present our
final results for $f_+(q^2)$ and $f_0(q^2)$, including statistical and
systematic errors, as coefficients of a series in the variable $z$ and the
covariance matrix between these coefficients. We then fit the lattice
form-factor data jointly with the experimentally measured differential decay
rate from BaBar to determine the CKM matrix element, $|V_{cb}|=(39.6 \pm
1.7_{\rm QCD+exp} \pm 0.2_{\rm QED})\times 10^{-3}$. As a byproduct of the
joint fit we obtain the form factors with improved precision at large recoil.
Finally, we use them to update our calculation of the ratio $R(D)$ in the
Standard Model, which yields $R(D) = 0.299(11)$.
