  An artin algebra $A$ is said to be CM-finite if there are only finitely many,
up to isomorphisms, indecomposable finitely generated Gorenstein-projective
$A$-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if
and only if every its Gorenstein-projective module is a direct sum of finitely
generated Gorenstein-projective modules. This is an analogue of Auslander's
theorem on algebras of finite representation type (\cite{A,A1}).
