A plane, gravity-driven, steady flow of a polar ice sheet over a horizontal bedrock, with no-slip basal conditions, is considered. The ice is modelled as a linearly viscous, incompressible and anisotropic fluid, with evolving orthotropic fabric depending on local strain rates and deformations. For prescribed free-surface elevation and non-uniform temperature field, the ice velocities required to maintain the assumed geometry are calculated by using the finite-element method. The focus is on the mechanism of dynamic (migration) recrystallization occurring near the base ofan ice sheet and leading to significant weakening, and ultimately to the complete loss, of the anisotropic fabric developed in the upper part of the ice cap. The weakening of the fabric with increasing strain rate and temperature is modelled by means of a scaling function depending continuously on a single effective strain-rate invariant. The results of numerical calculations demonstrate the effect of the recrystallization process on the overall flow rate of the ice sheet, and normalized velocity and strain-rate depth profiles are compared for flows with and without recrystallization involved to illustrate the effect of this mechanism on the large-scale behaviour of polar ice sheets.