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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. 2020 Mathematics Subject Classification: 30E05, 44A60, 46E10.doi:10.4064/sm190627-8-10 fatcat:xxphx5vpafhjlbfi46hki3w3im