  We characterize the surjectivity and the existence of a continuous linear
right inverse of the Stieltjes moment mapping on Gelfand-Shilov spaces, both of
Beurling and Roumieu type, in terms of their defining weight sequence. As a
corollary, we obtain some new results about the Borel-Ritt problem in spaces of
ultraholomorphic functions on the upper half-plane.
