The electrokinetic flow and attended electric conduction of a salt-free fluid, which contains counterions only, in a narrow slit channel subjected to a pressure gradient and an electric field were analytically investigated. The surface charge densities of the two plane walls of the slit are uniform but can be unequal. The electric potential and fluid velocity profiles were determined by solving the exact and linearized Poisson-Boltzmann equations and modified Navier-Stokes equation, respectively. Explicit formulas were obtained for the electroosmotic velocity and electric conductivity as functions of a dimensionless surface charge density and the ratio of surface charge densities of the slit walls. The relative surface potentials, average electroosmotic velocity, and average electric conductivity grow monotonically with an increase in the dimensionless surface charge density (or ratio of the half thickness of the slit to the Debye screening length) for a given value of the surface charge density ratio. But, when the surface charge density is high, the growths of the relative surface potentials and average electroosmotic velocity with its increase are suppressed noticeably owing to the effect of counterion condensation. For a salt-free solution in a planar slit with two equally charged walls, its relative surface potential and average electroosmotic velocity are greater but its average electric conductivity is smaller than the corresponding results in a circular tube provided that the thickness of the slit equals the diameter of the tube.