A vertically aligned spiral spring of negligible mass, fixed at its lower end, and carrying a heavy load which is constrained to move in the vertical direction, the motion being resisted by viscous damping, is considered. For a real spring, a large compression causes the coils to close up and ultimately to form an impenetrable barrier to the load. Thus, the spring control force becomes infinite at a certain point, and the non-linear differential equation of motion has a singularity at this point, in contrast to the classical cases of mass-spring systems. Assuming plausible analytic expressions for the singular control force, we obtain several exact solutions of the differential equation. The effects of the magnitudes of the damping, the control and the constant terms of the differential equation on the character of the motion (over-damped or oscillatory) are exhibited for several concrete cases that may be useful in applications.