We study the class of integrally closed domains having a unique
Kronecker function ring, or equivalently, domains in which the
$b$-operation is the only e.a.b star operation of finite type, as an
approach to domains having a quite simple set of valuation overrings.
We give characterizations by means of valuation overrings and integral
closure of finitely generated ideals. We provide new examples of such
domains and show that for several well-known classes of integral
domains the property of having a unique Kronecker function ring makes
them fall into the class of Pruefer domains.