The buoyancy-driven magnetohydrodynamic flow in a liquid-metal filled cubic enclosure was investigated by three-dimensional numerical simulation. The enclosure was differentially heated at two opposite vertical walls, all other walls being adiabatic, and a uniform magnetic field was applied orthogonal to the temperature gradient and to the gravity vector. The Rayleigh number was 10(5) and the Prandtl number was 0.0321 (characteristic of Pb-17Li at 573 K). The Hartmann number was made to vary between 10(2) and 10(3) and the electrical conductance of the walls between 0 and infinity. The continuity, momentum and enthalpy transport equations. in conjunction with a Poisson equation for the electric potential, were solved by a finite volume method using the general-purpose CFX-4 package with some necessary adaptations. Steady-state conditions were assumed. With respect to the case of parallel flow in an infinitely tall enclosure, studied in previous work, the suppression of convective motions due to magnetohydrodynamic interactions was stronger in the core, and a complex three-dimensional flow (with secondary motions) and current pattern was established in the fluid domain, Increasing the Hartmann number suppressed convective motions and exalted the square-shape of the circulation cells. Increasing the wall conductance ratio from perfectly insulating to perfectly conducting walls also resulted in an increasing suppression of convection. The related case of an internally heated enclosure is discussed in a companion paper.