An exact solution to the problem of collisionless, space‐charge‐limited flow of cold ions across a one‐dimensional (planar) dc plasma sheath of negligible electron density is derived for general values of the presheath ion velocityv0and electric fieldE0. For a given ion current densityJand sheath thicknessd, the exact solution reduces to the classical Child–Langmuir model in the case thatv0=0 andE0=0. When eitherv0orE0is sufficiently large, however, the exact solution may differ appreciably from the Child–Langmuir law. The existence of a closed‐form expression for the spatial variation of the sheath potential is shown to be contingent upon the satisfaction of a simple inequality relatingv0andE0toJ. Whenv0obeys the Bohm criterion and the magnitude ofE0suggests that the Bohm energy is acquired over a distance not less than one Debye length, this inequality is indeed satisfied.