Three velocity components of a Newtonian and a weakly elastic shear-thinning non-Newtonian fluid have been measured in an annulus with an eccentricity of 0.5, a diameter ratio of 0.5, and an inner cylinder rotation of 300 rpm. The results show that the rotation had similar effects on the Newtonian and non-Newtonian fluids, with a more uniform axial flow across the annulus and the maximum tangential velocities in the narrowest gap in both cases. The secondary flow circulation with the Newtonian fluid at a Reynolds number of 26,600 was in the direction of the rotation, with maximum values of 14% of the bulk velocity close to the inner pipe. With the 0.2% CMC polymer solution in laminar flow, rotation caused a narrow counter-rotating flow along the outer pipe wall, which was absent at a Reynolds number 9200. The turbulence intensities in the region of widest gap were uninfluenced by rotation, increased in the Newtonian fluid, and decreased in the non-Newtonian fluid in the region of the smallest gap. The flow resistance of both fluids increased with rotation at low Reynolds numbers and reduced with increasing values to become similar to those of nonrotating flows. Comparison between rotating results of the Newtonian and non-Newtonian fluids at a Reynolds number 9200 and the same inner cylinder rotation, showed effects similar to those of nonrotating flow with extension of nonturbulent flow, large reduction in turbulence intensities and drag reduction of the order of 61% for the CMC solution. The swirl velocities in both fluids were similar when the Rossby numbers were similar.