  The $\tau$ decay rate into hadrons of invariant mass smaller than
$\sqrt{s_0}\gg\Lambda_{\rm QCD}$ can be calculated in QCD assuming global
quark--hadron duality. It is shown that this assumption holds for
$s_0>0.7$~GeV$^2$. From measurements of the hadronic mass distribution, the
running coupling constant $\alpha_s(s_0)$ is extracted in the range
0.7~GeV$^2<s_0<m_\tau^2$. At $s_0=m_\tau^2$, the result is
$\alpha_s(m_\tau^2)=0.329\pm 0.030$. The running of $\alpha_s$ is in good
agreement with the QCD prediction.
