In a classical paper Plesset has determined conditions under which a bubble changing in volume maintains a spherical shape. The stability analysis was further developed by Prosperetti to include the effects of liquid viscosity on the evolving shape modes. In the present work the theory is further modified to include the changing density of the bubble contents. The latter is found to be important in violent collapses where the densities of the gas and vapor within a bubble may approach densities of the liquid outside. This exerts a stabilizing influence on the Rayleigh–Taylor mechanism of shape instability of spherical bubbles. A comparison with experimental data shows good agreement with the new theory; the Rayleigh–Taylor instability does provide an extinction threshold for violently collapsing bubbles. It is also explained why earlier works did not produce a slope in the Rayleigh–Taylor stability curve that conforms with that of the present work.