  We investigate the prospects of the search for a nonzero $\tau$ EDM form
factor $d_\tau(s)$ in $\tau$ pair production by $e^+ e^-$ collisions at BEPC-II
collider energies.
  We compute the expectation values and covariances of simple and optimal
$CP$-odd observables for $\tau$-pair production at $\sqrt{s}=4.18$ GeV and
$4.95$ GeV with subsequent decays of $\tau^\pm$ into major leptonic or
semihadronic modes. For the $\tau$ decays to two pions and three charged pions
we take the full kinematic information of the hadronic system into account.
  Applying cuts and using realistic assumptions on the eventually attainable
integrated luminosities at these energies, ${\cal L}(4.18) = 3\times 10^4~{\rm
pb}^{-1}$ and ${\cal L}(4.95) = 10^4~{\rm pb}^{-1}$, respectively, we find the
following.
  By taking into account purely semihadronic and semihadronic-leptonic
$\tau^+\tau^-$ decays one can achieve with optimal $CP$-odd observables the 1
s.d. sensitivities $\delta {\rm Re} d_\tau = 4.5\times 10^{-18}$ ecm
($5.3\times 10^{-18}$ ecm) and $\delta {\rm Im} d_\tau = 2.2 \times 10^{-18}$
ecm ($2.7\times 10^{-18}$ ecm) at $\sqrt{s}=4.18$ GeV ($4.95$ GeV).
