In this paper, we study the transverse vortex-induced vibrations of a cylinder with no structural restoring force (k = 0). In terms of the conventionally used normalized flow velocity, U*, the present experiments correspond to an infinite value (where U* = U/fND, fN = natural frequency, D = diameter). A reduction of mass ratios m* (mass/displaced mass) from the classically studied values of order m* = 100, down to m* = 1, yields negligible oscillations. However, a further reduction in mass exhibits a surprising result: large-amplitude vigorous vibrations suddenly appear for values of mass less than a critical mass ratio, m*crit = 0.54. The classical assumption, since the work of den Hartog (1934), has been that resonant large-amplitude oscillations exist only over a narrow range of velocities, around U*∼5, where the vortex shedding frequency is comparable with the natural frequency. However, in the present study, we demonstrate that, so long as the body’s mass is below this critical value, the regime of normalized velocities (U*) for resonant oscillations is infinitely wide, beginning at around U*∼5 and extending to U*→∞. This result is in precise accordance with the predictions put forward by Govardhan & Williamson (2000), based on elastically mounted vibration studies (where k > 0). We deduce a condition under which this unusual concept of an infinitely wide regime of resonance will occur in any generic vortex-induced vibration system.