  Background. The $^{12}$C$(\alpha,\gamma)^{16}$O reaction plays a fundamental
role in astrophysics and needs to be known with accuracy better than 10%.
Cascade $\gamma$ transitions through the excited states of $^{16}$O are
contributing to the uncertainty.
  Purpose. To constrain the contribution of the 0$^+$ (6.05 MeV) and 3$^-$
(6.13 MeV) cascade transitions.
  Method. The $\alpha$ Asymptotic Normalization Coefficient for these states
were measured using the $\alpha$-transfer reaction $^{12}$C$(^6$Li,$d)^{16}$O
at sub-Coulomb energies.
  Results. The 0$^+$ and 3$^-$ cascade transitions contribution was found to be
1.96$\pm$0.3 keV b and 0.12$\pm$0.04 keV b for destructive interference of the
direct and resonance capture and 4.36$\pm$0.45 keV b and 1.44$\pm$0.12 keV b
for constructive interference respectively.
  Conclusions. The combined contribution of the 0$^+$ and 3$^-$ cascade
transitions to the reaction at 300 keV does not exceed 4%. Significant
uncertainties have been dramatically reduced.
