In this paper, the stability of the energy balance in a fusion plasma is studied. We consider a uniform plasma, confined by a uniform magnetic field that does not penetrate the plasma. There is no energy input from external sources to the plasma or to the field coils, and no resistive losses in the coils. All species are assumed to have Maxwellian distributions at a common temperature.We write a set of particle-conservation equations, one for each ionic species, and two energy-balance equations, one for the plasma and one for the plasma plus the field. Thus the energy balance is described by a set of non-linear first-order differential equations. For steady state, these become algebraic equations, which can be solved numerically to determine the equilibrium states of the plasma.For small departures from equilibrium, the differential equations can be linearized. The linearized equations can be solved by the Laplace transform, and a stability condition can be obtained. However, this procedure involves some rather complicatei algebra.We find it more profitable to integrate the non-linear equations numerically, starting from equilibrium. We find that the energy balance is unstable at low plasma temperatures – where the highest power densities can be obtained. However, for reasonable plasma densities, the initial growth rate of the instability is very slow. Stabilization by means of some sort of negative-feedback system should be possible.