  The precise knowledge of the onset of the cross section $\sigma(e^+e^- \to
\tau^+\tau^-)$ at the threshold is necessary for improving the accuracy of
determination of the $\tau$ mass from the threshold measurements. The QED
radiative corrections of relative order $\alpha$ and $\alpha^2$, additional to
the well known Coulomb factor, are considered in the threshold region. The
correction terms of order $\alpha^2$ are calculated, which contain coefficients
enhanced by large parameters. As a result it is argued that the known
$O(\alpha)$ corrections provide the accuracy of the description of the cross
section close to $10^{-4}$, rather than $10^{-3}$ as claimed in a recent
literature. Also analytical expressions are provided for some limiting cases of
the corrections, previously calculated numerically.
