Stable affine models for algebraic group actions

Let G be a reductive linear algebraic group defined over an algebraically closed base field k of characteristic zero. A G-variety is an algebraic variety with a regular action of G, defined over k. An affine G-variety is called stable if its points in general position have closed G-orbits. We give a simple necessary and sufficient condition for a G-variety to have a stable affine birational model.