Steady natural convection in very low Prandtl number fluids is investigated using a double perturbation expansion in powers of the Grashof and Prandtl numbers. The fluid is contained in a horizontal cylindrical annulus, the walls of which either are held at constant temperature or support a constant heat flux. In both cases the evolution of the flow for increasing Grashof number is of interest. It is found that the basic flow pattern consists of one eddy. For both boundary conditions the center of this eddy falls into the lower half of the annulus as the Grashof number increases. Such behavior is directly opposite to experimental results obtained in fluids of higher Prandtl number.