A viscosity profile across the entire fluid outer core is found by interpolating between measured boundary values, using a differential form of the Arrhenius law governing pressure and temperature dependence. The discovery that both the retrograde and prograde free core nutations are in free decay (Palmer and Smylie, 2005) allows direct measures of viscosity at the top of the outer core, while the reduction in the rotational splitting of the two equatorial translational modes of the inner core allows it to be measured at the bottom. We find 2,371 plus/minus 1,530 Pa.s at the top and 1.247 plus/minus 0.035 x 10^11 Pa.s at the bottom. Following Brazhkin (1998) and Brazhkin and Lyapin (2000) who get 10^2 Pa.s at the top, 10^11 Pa.s at the bottom, by an Arrhenius extrapolation of laboratory experiments, we use a differential form of the Arrhenius law to interpolate along the melting temperature curve to find a viscosity profile across the outer core. We find the variation to be closely log-linear between the measured boundary values. The close agreement of the boundary values of viscosity, found by Arrhenius extrapolation of laboratory experiments, with those found from the free core nutations, and the inner core translational modes, suggests that core flows are laminar and that the returned viscosities are measures of their molecular values. This would not be the case in the presence of the vigorous turbulent convection sometimes postulated by dynamo theorists. The local Ekman number is found to range from 10^-2 at the bottom of the outer core to 10^-10 at the top. Except in the very lower part of the outer core, Ekman numbers are in the range 10^-4 to 10^-5, or below, in which the laminar flows of numerical dynamos and laboratory rotating fluids experiments occur.