It is demonstrated that the signal-to-noise ratio for a weak periodic signal in a superconductive loop with a Josephson junction (a superconducting quantum interference device, or SQUID) can be substantially enhanced, over a wide range of frequencies, by the addition of noise. This manifestation of zero-dispersion stochastic resonance (ZDSR) is shown to occur for a wide variety of loop parameters and signal frequencies. Unlike most earlier examples of stochastic resonance, ZDSR does not depend on fluctuational transitions between coexisting stable states. Rather, it exploits the noise-enhanced susceptibility that arises in underdamped nonlinear oscillators for which the oscillation eigenfrequency possesses one or more extrema as a function of energy. The phenomenon is investigated theoretically, and by means of analog and digital simulations. It is suggested that ZDSR could be used to enhance the sensitivity of radio-frequency SQUIDs and other SQUID-based devices. In the course of the work, two additional useful results were obtained: (a) an asymptotic expression describing ZDSR for the general case in the limit of weak dissipation; (b) a method for the numerical calculation of fluctuation spectra in bistable or multistable underdamped systems.