Algebraic group actions on normal varieties

Let G be a connected algebraic k-group acting on a normal k-variety X, where k is a field. We show that X is covered by open G-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a G-linearized vector bundle on an abelian variety, quotient of G. This generalizes a result of Sumihiro on actions of affine algebraic groups, and yields a refinement of a result of Weil on birational actions.