  The $^{13}{\rm C}(\alpha,n)^{16}{\rm O}$ reaction is considered to be the
main neutron source responsible for the production of heavy nuclides (from
${\rm Sr}$ to ${\rm Bi}$) through slow $n$-capture nucleosynthesis
($s$-process) at low temperatures during the asymptotic giant branch (AGB)
phase of low mass stars ($\lesssim 3-4\;{\rm M}_{\odot}$, or LMSs). In recent
years, several direct and indirect measurements have been carried out to
determine the cross section at the energies of astrophysical interest (around
$190\pm40\;{\rm keV}$). However, they yield inconsistent results causing a
highly uncertain reaction rate and affecting the neutron release in LMSs. In
this work we have combined two indirect approaches, the asymptotic
normalization coefficient (or ANC) and the Trojan Horse Method (THM), to
unambiguously determine the absolute value of the $^{13}{\rm
C}(\alpha,n)^{16}{\rm O}$ astrophysical factor. Therefore, we have determined a
very accurate reaction rate to be introduced into astrophysical models of
$s$-process nucleosynthesis in LMSs. Calculations using such recommended rate
have shown limited variations in the production of those neutron-rich nuclei
(with $86\leq A\leq 209$) receiving contribution only by slow neutron captures.
