  In the QCD Sum Rule determination of $m_s$ using the two-point correlator of
divergences of $\Delta S=1$ vector currents, the final uncertainty on $m_s$ is
mainly due to the hadronic spectral function. Using a specific parameterization
which fully takes into account the available experimental data on the $K \pi$
($I=1/2, J^P=0^+)$ system, characterized by the presence of a relevant
nonresonant component in addition to the resonant one, we find ${\bar m_s}(1
GeV)\ge 120 MeV$. In particular, varying only the parameters describing the
nonresonant $K \pi$ component and $\Lambda_{\bar {MS}}^{n_f=3}$ we obtain
${\bar m_s}(1 GeV)=125 - 160 MeV$. This result is smaller than analogous ones
obtained by using a parameterization in terms of only resonant states. We
discuss how to systematically improve the determination of $m_s$ by this
method.
