An analytical study is presented of the steady electrokinetic flow in a long uniform circular capillary bearing a layer of adsorbed polyelectrolytes on its inside wall. In this solvent-permeable and ion-penetrable surface charge layer, idealized polyelectrolyte segments are assumed to distribute at a uniform density. The electrical potential and space charge density distributions on a cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes equation. Explicit formulas for the volumetric flow rate, the electro-osmotic velocity, the electric current and the streaming potential in the capillary are also derived. The results demonstrate that the structure of the surface charge layer can result in an augmented or a diminished electrokinetic flow relative to that in a capillary with bare walls, depending on the characteristics of the electrolyte solution, of the surface charge layer, and of the capillary. The Onsager reciprocal relation is found to be satisfied in the general case.