  We investigate the subleading-power corrections to the exclusive $B\to D \ell
\nu_\ell$ form factors at ${\cal O} (\alpha_s^0)$ in the light-cone sum rules
(LCSR) framework by including the two- and three-particle higher-twist
contributions from the $B$-meson light-cone distribution amplitudes up to the
twist-six accuracy, by taking into account the subleading terms in expanding
the hard-collinear charm-quark propagator, and by evaluating the hadronic
matrix element of the subleading effective current $\bar q \, \gamma_{\mu} \, i
\not\!\!{D}_\perp / (2 \, m_b) \, h_v$. Employing further the available
leading-power results for the semileptonic $B \to D$ form factors at the
next-to-leading-logarithmic level and combining our improved LCSR predictions
with the recent lattice determinations, we then carry out a comprehensive
phenomenological analysis on the semi-leptonic $B\to D \ell \nu_\ell$ decay. We
extract $|V_{cb}| = \big( 40.2^{+0.6}_{-0.5} {\big |_{\rm th}}\,\,
{}^{+1.4}_{-1.4} {\big |_{\rm exp}} \big)\times 10^{-3}$ ($|V_{cb}| = \big(
40.9^{+0.6}_{-0.5} {\big |_{\rm th}}\,\, {}^{+1.0}_{-1.0} {\big |_{\rm exp}}
\big)\times 10^{-3}$) using the BaBar (Belle) experimental data, and
particularly obtain for the gold-plated ratio $R(D) = 0.302\pm 0.003$.
