  Certain CP-odd momentum correlations in the production and subsequent decay
of $\tau$ pairs in $e^+ e^-$ collisions are enhanced significantly when the
$e^+$ and $e^-$ beams are longitudinally polarized. These may be used to probe
the real and imaginary parts of $d_\tau^\gamma$, the electric dipole moment of
the $\tau$. Closed-form expressions for these ``vector correlations'' and the
standard deviation of the operators defining them due to standard model
interactions are presented for the two-body final states of $\tau$ decays. If
42\% average polarization of each beam is achieved, as proposed for the
tau-charm factories, with equal integrated luminosities for each sign of
polarization and a total yield of $2\cdot 10^7$ $\tau^+ \tau^-$ pairs, it is
possible to attain sensitivities for $\vert\delta {\rm Re}
d_{\tau}^{\gamma}\vert$ of $8\cdot 10^{-19}$, $1\cdot 10^{-19}$, $1\cdot
10^{-19}$ $e$ cm respectively and for $\vert\delta {\rm Im}
d_{\tau}^{\gamma}\vert$ of $4\cdot 10^{-14}$, $6\cdot 10^{-15}$, $5\cdot
10^{-16}$ $e$ cm respectively at the three operating center-of-mass energies of
3.67, 4.25 and 10.58 GeV. These bounds emerge when the effects of a posible
weak dipole form factor $d_\tau^Z$ are negligible as is the case when it is of
the same order of magnitude as $d_\tau^\gamma$. Furthermore, in such a
polarization experiment where different polarizations are possible, a
model-independent disentangling of their individual effects is possible, and a
technique to achieve this is described. A strong longitudinal polarization
physics programme at the tau-charm factory appears warranted.
