  We show that a smooth projective non-degenerate arithmetically Cohen-Macaulay
subvariety X of P^N infinite Cohen-Macaulay type becomes of finite
Cohen-Macaulay type by removing Ulrich bundles if and only if N = 5 and X is a
quartic scroll or the Segre product of a line and a plane. In turn, we give a
complete and explicit classification of ACM bundles over these varieties.
