The tau weak-magnetic dipole moment

We calculate the prediction for the anomalous weak-magnetic form factor of the tau lepton at $q^2=M_Z^2$ within the Standard Model. With all particles on-shell, this is a electroweak gauge invariant quantity. Its value is $a_\tau^w (M_Z^2)= - \;(2.10 + 0.61\, i) \times 10^{-6}$. We show that the transverse and normal components of the single-tau polarization of tau pairs produced at $e^+e^-$ unpolarized collisions are sensitive to the real and absorptive parts of the anomalous weak-magnetic dipole moment of the tau. The sensitivity one can achieve at LEP in the measurement of this dipole moment is discussed.