The motion of a heavy ion in the presence of an intense ultralow-frequency electromagnetic wave propagating along the dc magnetic field is analyzed. Starting from the basic equations of motion and from their associated two invariants, the heavy ion velocity-space trajectories are drawn. It is shown that after a certain time, particles whose initial phase angles are randomly distributed tend to bunch together, provided that the wave intensity, b1 is sufficiently large. The maximum perpendicular velocity va which the particle reaches during the bunching process can be a fraction of the Alfv¿n velocity. When the wave frequency f is smaller than the heavy ion gyrofrequency, F, va is maximum for waves with frequencies slightly smaller than F. When the wave frequency is larger than the cutoff frequency fco, fa is a continuously decreasing function of f. In both cases, va is proportional to b1/ ‖ F-f ‖ when b1 is small and/or ‖ F-f ‖ is large, and it is proportional to (b1/f)1/3 when b1 is larger and/or ‖ F-f ‖ is small. The process is nonresonant in the sense that it is efficient however small is the heavy ion initial parallel velocity. A parameter study is done which gives the process efficiency for different wave or particle characteristics. Approximate analytical expressions are given, and they are compared with the results of numerical computations. The importance of these results for the interpretation of the recently observed acceleration of He+ ions in conjunction with the occurrence of large-amplitude ion cyclotron waves in the equatorial magnetosphere is discussed.