We rapidly change the scattering length a(s) of a Rb-87 Bose-Einstein condensate by means of a Feshbach resonance, simultaneously releasing the condensate from its harmonic trapping potential. When a(s) is changed from positive to negative, the subsequent collapse of the condensate is stabilized by the kinetic energy imparted during the release, resulting in a deceleration of the loss rate near the resonance. We also observe an increase in the Thomas-Fermi radius, near the resonance, that cannot be understood in terms of a simple scaling model. Instead, we describe this behavior using the Gross-Pitaevskii equation, including three-body recombination, and hypothesize that the increase in cloud radius is due to self-interference of the condensate resulting in the formation of concentric shells.