On the stability of the action of an algebraic group on an algebraic variety

We prove the following fact: if a connected algebraic group having no rational characters acts regularly on a normal irreducible algebraic variety $X$ with periodic divisor class group $ClX$, then for the orbit $O_x$ of a point $x\in X$ in general position to be closed, it is sufficient that $O_x$ be an affine variety; moreover, if $X$ is affine, this condition is also sufficient.