LetXXbe a smooth projective Berkovich space over a complete discrete valuation fieldKKof residue characteristic zero, and assume thatXXis defined over a function field admittingKKas a completion. Let furtherμ\mube a positive measure onXXandLLbe an ample line bundle such that the mass ofμ\muis equal to the degree ofLL. We prove the existence of a continuous semipositive metric whose associated measure is equal toμ\muin the sense of Zhang and Chambert-Loir. We do this under a technical assumption on the support ofμ\mu, which is, for instance, fulfilled if the support is a finite set of divisorial points. Our method draws on analogs of the variational approach developed to solve complex Monge-Ampère equations on compact Kähler manifolds by Berman, Guedj, Zeriahi, and the first named author, and of Kołodziej’sC0C^0-estimates. It relies in a crucial way on the compactness properties of singular semipositive metrics, as defined and studied in a companion article.