The importance of the stability problem for matrices of the special form called mammillary matrices has been noted by Hearon (1). The stability problem may be formulated in terms of the Liapounov matrix equation. A constructive procedure, dependent upon the particular structure of mammillary matrices, for the solution of the Liapounov equation and the subsequent solution of the stability problem is described in this note. For an n x n matrix the principal computational steps are the solution of an n x n linear system and the determination of the rank and signature of an n x n symmetric matrix.