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Let $F$ be a field, and let Zar$(F)$ be the space of valuation rings of $F$with respect to the Zariski topology. We prove that if $X$ is a quasicompactset of rank one valuation rings in Zar$(F)$ whose maximal ideals do notintersect to $0$, then the intersection of the rings in $X$ is an integraldomain with quotient field $F$ such that every finitely generated ideal is aprincipal ideal. To prove this result, we develop a duality between (a)quasicompact sets of rank one valuation rings whose maximal ideals do notintersect to $0$, and (b) one-dimensional Pr\"ufer domains with nonzeroJacobson radical and quotient field $F$. The necessary restriction in all thesecases to collections of valuation rings whose maximal ideals do not intersectto $0$ is motivated by settings in which the valuation rings considered alldominate a given local ring.

A principal ideal theorem for compact sets of rank one valuation rings