A revised theory is presented of slit‐ and capillary‐die rheometry, which, contrary to the earlier ones, predicts that the ratio of the second normal stress difference to the first, (S rr −S θθ )/(S zz −S rr ), is negative in sign and its magnitudes is always less than unity. In order to test the theory, use is made of exit‐pressure measurements for five polymer melts and six polymer solutions. Exit pressures of polymer melts were obtained from both slit and capillary dies, and exit pressures of polymer solutions were obtained from slit dies with flush‐mounted pressure transducers. The values of first normal stress differences determined from exit‐pressure measurements show good agreement with the ones measured with a Weissenberg rheogoniometer. It has been found that (S rr −S θθ )/(S zz —S rr ) lies between −0.4 and −0.6 under the conditions investigated for five polymer melts. This is in line with recent measurements of the normal stress ratio of various polymer solutions.