The purpose of this note is to prove that under certain hypotheses, the graded ring of integral automorphic forms, with respect to an arithmetic group operating On a tube domain, is generated as a graded algebra over the complex numbers by a finite number of automorphic forms having rational integral Fourier coefficients. The result, which seems to have some potential numbertheoretic interest, was inspired by notes of M. Eichler [8, esp. Satz 49] and by questions raised in correspondence with that author and with I. I. Pyatetskii-Shapiro. One should also note the similarity of ideas here with those used in the proof of Satz D of [9]. We wish to remark, furthermore, that in most cases lengthy computations will be needed to verify the applicability of our theorem here, which should therefore be regarded more as a technical lemma than as a substantial contribution in itself. As in [2], let ~ denote a (hermitian) symmetric tube domain, let F be a discrete, arithmetically defined subgroup of the group of all holomorphic automorphisms of ~ , and assume that ~ has a zerodimensional rational boundary component Fo, which we take to be the zero-dimensional rational boundary component of ~ at infinity as in [2]. Let