The buoyancy-driven magnetohydrodynamic flow in a liquid-metal filled cubic enclosure with internal heat generation was investigated by three-dimensional numerical simulation. The enclosure was volumetrically heated by a uniform power density and was cooled along two opposite vertical walls, all other walls being adiabatic. A uniform magnetic field was applied orthogonally to the gravity vector and to the temperature gradient (i.e., parallel to the isothermal walls). The-Prandtl number was 0.0321 (characteristic of Pb-17Li at 573 K); the Rayleigh number was made to vary from 10(5) to 10(7), the Hartmann number between 10(2) and 10(3) and the electrical conductance of the walls between 0 and infinity. The Navier-Stokes equations. in Conjunction with a scalar transport equation for the fluid's enthalpy and with the Poisson equation for the electrical potential. were solved by a finite volume method using the CFD package CFX-4 with some necessary adaptations. Steady-state conditions were assumed. In all cases, a three-dimensional flow with complex secondary motions and a complex current pattern was established. The effects of Hartmann number, wall conductance ratio and Rayleigh number were discussed and results were compared with those previously obtained for fully developed flow in an infinitely tall, internally heated channel of square cross-section. The related case of a differentially heated cubic enclosure is discussed in a companion paper.