  We show that transverse and normal single-$tau$ polarization of $tau$ pairs
produced at $e^+$ $e^-$ unpolarized collisions, at the $Z$ peak, are sensitive
to weak (magnetic and electric) dipole moments of the $tau$. We also show how
these components of the $\tau$ polarization are accessible by measuring
appropriate azimuthal asymmetries in the angular distribution of its decay
products. Sensitivities of the order of $10^{-18}$ $e\cdot cm$, for the
weak-electric dipole moment, and $10^{-4}$ ($10^{-3}$), for the real
(imaginary) part of the weak-magnetic dipole moment of $\tau$, may be achieved.
Compatible bounds are also presented from spin-spin correlated asymmetries.
