A plane, gravity-driven, steady flow of an isothermal 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 that depends on local strain rates and deformations. For a fixed, free-surface elevation, the ice-accumulation rates necessary to maintain the prescribed geometry are calculated by using the finite-element method, together with the velocities and stresses. Numerical simulations have been carried out for different combinations of enhancement factors for compression and shear in order to investigate their effect on the rate of flow. The results obtained have shown that, apart from the near-divide region, the global flow rate is nearly proportional to the magnitude of the shear-enhancement factor and is very little sensitive to the value of the compression enhancement factor. Normalized velocity-depth profiles have been compared for the anisotropic and isotropic ice and it has been found that significant differences occur only in a region near the ice divide. Direct shear stresses are little affected by the ice anisotropy, but the longitudinal deviatoric stresses in a part of the ice sheet are significantly increased compared to the isotropic ice flow.