  Exclusive semileptonic B decays into $D_1$ and $D_2^*$ mesons are
investigated including order $\Lambda_{QCD}/m_{c,b}$ corrections using the
heavy quark effective theory. At zero recoil, the $\Lambda_{QCD}/m_{c,b}$
corrections can be written in terms of the leading Isgur-Wise function for
these transitions, $\tau$, and known meson mass splittings. We obtain an almost
model independent prediction for the shape of the spectrum near zero recoil,
including order $\Lambda_{QCD}/m_{c,b}$ corrections. We determine $\tau(1)$
from the measured $B\to D_1\ell\bar\nu$ branching ratio. Implications for B
decay sum rules are discussed.
