  In this paper we prove that, for every $r \geq 2$, the moduli space
$M^s_X(r;c_1,c_2)$ of rank $r$ stable vector bundles with Chern classes
$c_1=rH$ and $c_2=(3r^2-r)/2$ on a nonsingular cubic surface $X \subset
\mathbb{P}^3$ contains a nonempty smooth open subset formed by ACM bundles,
i.e. vector bundles with no intermediate cohomology. The bundles we consider
for this study are extremal for the number of generators of the corresponding
module (these are known as Ulrich bundles), so we also prove the existence of
indecomposable Ulrich bundles of arbitrarily high rank on $X$.

Algebraic Geometry;Commutative Algebra