  We perform the first calculation of form factors in the semileptonic decays
$B\!\to\! D_1(2420)\ell\nu_\ell$ and $B \to D_1^\prime (2430)\ell \nu_\ell$
using QCD light-cone sum rules (LCSRs) with $B$-meson distribution amplitudes.
In this calculation the $c$-quark mass is finite. Analytical expressions for
two-particle contributions up to twist four are obtained. To disentangle the
$D_1$ and $D_1^\prime$ contributions in the LCSRs, we suggest a novel approach
that introduces a combination of two interpolating currents for these charmed
mesons. To fix all the parameters in the LCSRs, we use the two-point QCD sum
rules for the decay constants of $D_1$ and $D_1^\prime$ mesons augmented by a
single experimental input, that is the $B \to D_1(2420)\ell\nu_\ell$ decay
width. We provide numerical results for all $B\to D_1$ and $B\to D_1^\prime$
form factors. As a byproduct, we also obtain the $D_1$- and $D_1'$-meson decay
constants and predict the lepton-flavour universality ratios $R(D_1)$ and
$R(D_1')$.
