In this paper, the flow of couple stress fluid is investigated in a helical screw rheometer (HSR). By unwrapping the channel, lands, and the outside rotating barrel, the geometry of HSR is approximated as a shallow infinite channel. Both one- and two-dimensional analysis of the problem is presented using rectangular coordinates. In either case an exact solution of the flow problem is presented and the formulas of velocity profile and volumetric flow rate are obtained as a function of couple stress parameter. It is observed that velocity profile decreases in going from Newtonian to couple stress fluid which indicates a decrease in extrusion process for a couple stress fluid in comparison with Newtonian fluid. Moreover, the volumetric flow rate is found to be a decreasing function of couple stress parameter.