  We show that small perturbations of the spatially homogeneous equilibrium of
a thermally driven compressible viscous fluid are globally stable.
Specifically, any weak solution of the evolutionary Navier--Stokes--Fourier
system driven by thermal convection converges to an equilibrium as time goes to
infinity. The main difficulty to overcome is the fact the problem does not
admit any obvious Lyapunov function. The result applies, in particular, to the
Rayleigh--B\' enard convection problem.
