  We prove that in characteristic zero the multiplication of sections of
dominant line bundles on a complete symmetric variety $X=\bar{G/H}$ is a
surjective map. As a consequence the cone defined by a complete linear system
over $X$, or over a closed $G$ stable subvariety of $X$ is normal. This gives
an affirmative answer to a question raised by Faltings. A crucial point of the
proof is a combinatorial property of root systems.
