Fully developed steady flow of a highly elastic liquid such as a molten polymer is not possible at low Reynolds number above a critical value of the elastic shear strain (wall shear stress/modulus). The limiting value is 2.6 in a cylindrical die and 0.7 in a slit die. The result is obtained using the linear theory of hydrodynamic stability and assuming that the fluid rheological properties are adequately described by the convected Maxwell model. Agreement with reported melt flow instabilities in extrusion is encouraging.