  A self-consistent mode-coupling theory is presented for the viscosity of
solutions of charged rod-like polymers. The static structure factor used in the
theory is obtained from polymer integral equation theory; the Debye-H\"{u}ckel
approximation is inadequate even at low concentrations. The theory predicts a
non-monotonic dependence of the reduced excess viscosity, $\eta_R$, on
concentration from the behaviour of the static structure factor in
polyelectrolyte solutions. The theory predicts that the peak in $\eta_R$ occurs
at concentrations slightly lower than the overlap threshold concentration,
$c^\ast$. The peak height increases dramatically with increasing molecular
weight and decreases with increased concentrations of added salt. The position
of the peak, as a function of concentration divided by $c^\ast$ is independent
of salt concentration or molecular weight. The predictions can be tested
experimentally.
