The steady electrokinetic flow of an electrolyte solution in a narrow capillary tube or slit generated by a uniform prescribed concentration gradient is analytically studied. The electric double layer adjacent to the charged capillary wall may have arbitrary thickness relative to the capillary radius. The electrostatic potential distribution on a cross section of the capillary is obtained by solving the linearized Poisson-Boltzmann equation, which applies to the case of low surface potential or low surface charge density at the capillary wall. Explicit formulas for the fluid velocity profile due to the gradient of electrolyte concentration through the capillary are derived as the solution of a modified Navier-Stokes equation. In the absence of a macroscopic electric field induced by the electrolyte gradient, the fluid flows (due to the chemiosmotic contribution) toward lower electrolyte concentration. With an induced electric field, competition between electroosmosis and chemiosmosis can result in more than one reversal in direction of the fluid now over a small range of the surface potential. For a given concentration gradient of electrolyte in a capillary, the fluid flow rate does not always increase with an increase in the electrokinetic radius of the capillary, which is the capillary radius divided by the Debye screening length.