EXACT solutions of the low mass transfer rate equations are obtained for the fully developed vapour mass fraction, temperature, velocity and pressure gradient in a vertical cavity subject to horizontal heat and mass transfer. The solutions for rectangular sections are in terms of Fourier series, while for elliptic sections a simple polynomial describes the vertical velocity. Uniqueness of the solutions is demonstrated under the restriction that the mass fraction, velocity and temperature fields are independent of height. For a cavity of rectangular section, it is predicted and verified that the flow in the plane of spanwise symmetry is practically independent of the span if this exceeds 1.7 times the width.