  We consider the Catalan equation $x^p - y^q = 1$ in unknowns $x, y, p, q$,
where $x, y$ are taken from an integral domain $A$ of characteristic $0$ that
is finitely generated as a $\mathbb{Z}$-algebra and $p, q > 1$ are integers. We
give explicit upper bounds for $p$ and $q$ in terms of the defining parameters
of $A$. Our main theorem is a more precise version of a result of Brindza.
Brindza also gave inexplicit bounds for $p$ and $q$ in the special case that
$A$ is the ring of $S$-integers for some number field $K$. As part of the proof
of our main theorem, we will give a less technical proof for this special case
with explicit upper bounds for $p$ and $q$.
