Measurements of the temperature dependence of the shear viscosity of the ionic binary mixture ethylammonium nitrate in n-octanol near the critical consolute point are reported. The kinematic viscosity is measured with a capillary viscometer in the range of reduced temperature 3X10(-5)less than or equal to t=(T-T-c)/T-c less than or equal to 6.2X10(-2). The temperature dependence of the density is measured by a volume dilatometer. No critical density anomaly is observed within the temperature range 5X10(-4)less than or equal to t less than or equal to 4X10(-2). Shear viscosity data are consistent with a power law divergence eta=eta(0) (Q xi(0))(-gamma/nu)t(-gamma)F predicted by the mode-coupling and the dynamic renormalization group theories. The value of the critical exponent gamma agrees with the theoretical value y=0.032 when the critical temperature T-c is fixed at the experimental value. When T-c is set as a free parameter the fits give 0.0385 less than or equal to gamma less than or equal to 0.0438, depending on the forms chosen for the background viscosity eta(0) and the crossover function F. Estimation of the wave number Q is found in agreement with data obtained for neutral fluids.