We report the apparent viscosity of aqueous solutions of a well-characterized synthetic polyelectrolyte, the sodium salt of polystyrene sulfonate. Using two rheometers we measure the apparent viscosity over more than five decades of shear rate to determine the Newtonian viscosity and the onset of shear thinning, which is inversely proportional to the relaxation time. We study five decades of polyelectrolyte concentration, from the dilute to the entangled regimes, and three decades of added salt (NaCl) concentration, on a single polyelectrolyte sample. Much of the viscosity data in the literature are shown to be in the shear thinning regime and are not indicative of the Newtonian viscosity. Our data for viscosity and longest relaxation time are qualitatively consistent with a recent scaling theory. There are systematic deviations from this simple theory, including the failure of the well-established Fuoss law, which is shown to be obeyed by the apparent viscosity only at higher shear rates.