A new device has been built and tested which provides for pressure‐driven flow at high shear rates in the angular direction within the annulus between two concentric cylinders. Analysis of this viscometric flow demonstrates that the measured difference between the normal stresses at the inner and outer surfaces of the annulus is related to the first normal stress function. The data interpretation corrects for pressure hole errors and inertial effects. First normal stress function data for aqueous polymeric solutions are presented and discussed. Significant advantages over commonly used testing procedures are realised. These include the ability to conduct tests on both polymer solutions and melts over a broad range of shear rates (comparable to that realized in capillary flows), improved temperature control, and the minimization of changes in the sample due to solvent evaporation and polymer degradation. Surmountable difficulties include the need for relatively large sample size, the possibility of secondary flows under some conditions, and data analysis that requires inversion of an integral.