  Background: Accurate knowledge of the $^{13}$C($\alpha$,$n$)$^{16}$O reaction
cross section is important for the understanding of the s-process in AGB stars,
since it is considered to be the main source of neutrons. The sub-threshold
1/2$^+$ state at excitation energy of 6.356 MeV in $^{17}$O has a strong
influence on the reaction cross section at energies relevant for astrophysics.
Several experiments have been performed to determine the contribution of this
state to the $^{13}$C($\alpha,n)^{16}$O reaction rate. Nevertheless,
significant discrepancies between different measurements remain.
  Purpose: The aim of this work is to investigate these discrepancies.
  Method: An 8 MeV $^{13}$C beam (below the Coulomb barrier) was used to study
the $\alpha$-transfer reaction $^6$Li($^{13}$C,$d$)$^{17}$O.
  Results: The squared Coulomb modified ANC of the 1/2$^+$ state in $^{17}$O
measured in this work is $(\tilde
C^{^{17}\text{O}(1/2+)}_{\alpha-^{13}\text{C}})^2=3.6\pm0.7\hspace{0.2cm}\text{fm}^{-1}$.
  Conclusions: Discrepancy between the results of $\alpha$-transfer experiments
have been resolved. However, some discrepancy with the most recent measurement
using the Trojan Horse method remains.
