  We study radiative corrections to $\bar{B} \to X_c \ell \bar{\nu}_\ell$
decays assuming the power counting $m_c \sim \sqrt{\Lambda_QCD m_b}$ for the
charm-quark mass. Concentrating on the shape-function region, we use effective
field-theory methods to calculate the hadronic tensor at NLO accuracy. From
this we deduce a shape-function independent relation between partially
integrated $\bar{B} \to X_c\ell\bar{\nu}_\ell$ and $\bar{B} \to X_u \ell
\bar{\nu}_\ell$ spectra to leading power in $1/m_b$, including first-order
corrections in the strong coupling constant. This may provide an independent
cross-check on the determination of the CKM element |V_{ub}|.
