The Morse normal form of linear systems is a fundamental result of broad interest in systems and control theory. The form is canonical relative to the group of state feedback, output injection, and input-, output-, and state-coordinate transformations. The complete system invariant under the action of this group consists of three lists of integers and one list of polynomials. Stability of the system, however, is not invariant under this action. In problems where stability matters, one needs a more specific result, the stability-preserving Morse normal form. This new form applies to stable systems, and it is canonical with respect to stability-preserving state feedback, stability-preserving output injection, and input-, output-, and state-coordinate transformations. The complete invariant is shown to consist of three lists of integers and two lists of polynomials, one having only stable zeros and the other one only unstable zeros. The canonical system representation consists of four subsystems three of which are ordered cascade realizations of prime building blocks, and the fourth one realizes a Jordan block matrix.