The attitude motion of a dual-spin spacecraft consisting primarily of two symmetric bodies with a common axis of rotation and a constant relative spin rate is analyzed. Each of the main bodies contains a spring-mass-dashpot damper with a degree of freedom parallel to their common axis. The equations of motion are studied by means of a perturbation analysis in which only the damper masses are assumed small. The problem then becomes the determination of perturbations to a reference motion which is similar to the general precession of the torque-free rigid body in classical mechanics. One of the key parameters associated with such motion is the nutation angle (or the angle between the nominal spin axis and the inertially-fixed angular momentum vector). By determining the zeroth and first order behavior of this angle for this system, a general analytical stability criterion is established. This criterion not only determines the stability or instability of an arbitrary initial state of the system, but reveals the nature of any instability which might exist. In reduced form, the stability criterion verifies results in the recent literature obtained by Routhian and Floquet analyses of linearized equations representing special cases of this system.