The problem of the finite amplitude, horizontal oscillatory motion of a mass attached to a neo-Hookean rubber spring and supported by a fixed, ideally smooth and rigid horizontal surface is solved exactly in terms of the Heuman lambda and beta functions. Therefore, the period of the oscillations may be computed from tables of values of the complete lambda function. It is proved that the ratio of the amplitude-dependent frequency of any finite amplitude motion to the constant frequency of the small amplitude vibration of the same oscillator depends only on the assigned initial data. Therefore, the ratio is universal for every neo-Hookean oscillator regardless of its stiffness or of its design parameters. Upper and lower bounds for this ratio also are provided. Moreover, it is proved for assigned initial data, that the normalized amplitude of the vibrations is invariable for every neo-Hookean oscillator, and all energy curves are reduced to a single constant energy trajectory determined by the initial data alone. The slingshot effect that occurs for a slender neo-Hookean rubber cord also is described, and all results are illustrated graphically.