A self-consistent mode-coupling theory is presented for the viscosity of solutions of charged rodlike polymers. The static structure factor used in the theory is obtained from polymer integral equation theory; the Debye-Huckel approximation is inadequate even at low concentrations. The theory predicts a nonmonotonic dependence of the reduced excess viscosity eta(R) on concentration from the behavior 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(*). 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(*), is independent of salt concentration or molecular weight. The predictions can be tested experimentally. (C) 2004 American Institute of Physics.