  We present an analytical calculation of the $\alpha_s$ corrections
  for the coefficient of $\rho_D/m_Q^3$ term in the heavy quark expansion
  for the inclusive semileptonic decays of heavy hadrons, such as
  $B \to X_c \ell \bar{\nu}$. The full dependence of the coefficient
  on the final-state quark mass is taken into account.
  Our result leads to further improvement of
  the theoretical predictions for the precision determination of
  CKM matrix element $|V_{cb}|$.
