  Let $V$ be a rank one valuation domain with quotient field $K$. We
characterize the subsets $S$ of $V$ for which the ring of integer-valued
polynomials ${\rm Int}(S,V)=\{f\in K[X] \mid f(S)\subseteq V\}$ is a Pr\"ufer
domain. The characterization is obtained by means of the notion of
pseudo-monotone sequence and pseudo-limit in the sense of Chabert, which
generalize the classical notions of pseudo-convergent sequence and pseudo-limit
by Ostrowski and Kaplansky, respectively. We show that ${\rm Int}(S,V)$ is
Pr\"ufer if and only if no element of the algebraic closure $\overline{K}$ of
$K$ is a pseudo-limit of a pseudo-monotone sequence contained in $S$, with
respect to some extension of $V$ to $\overline{K}$. This result expands a
recent result by Loper and Werner.
