  $\DeclareMathOperator{\IntR}{Int{}^\text{R}}$Integer-valued rational
functions are a natural generalization of integer-valued polynomials. Given a
domain $D$, the collection of all integer-valued rational functions over $D$
forms a ring extension $\IntR(D)$ of $D$. For a valuation domain $V$, we
characterize when $\IntR(V)$ is a Pr\"ufer domain and when $\IntR(V)$ is a
B\'ezout domain. We also extend the classification of when $\IntR(D)$ is a
Pr\"ufer domain.
