The suppression of macroscopic oscillations (so-called "sawteeth") of the central region of a magnetically confined plasma column is related to the effect of magnetically trapped energetic nuclei produced by the injection of rf waves or neutral beams. We evaluate the threshold for, and describe the process involved in, the transition of m0 = 1 modes, which trigger the crash phase of sawtooth oscillations, to a regime that is shown to be stable.