The constrained or constant radius magnetic acceleration of a strong non‐neutral proton ring by a modified betatron field is considered. The modified betatron field consists of an azimuthal magnetic fieldB&Vthgr;superimposed on a betatron fieldBzin which the 1 : 2 flux rule is satisfied. An important advantage of the constrained acceleration is that the energy of the nonrelativistic ions increases with the square of the magnetic field. It has been found that the orbits of the gyrating particles are stable when the self‐field indexns{=&ohgr;2p[2&ggr;0&Vthgr;(t) &OHgr;20(t)]−1} is much greater than unity, provided thatns< (B&Vthgr;/2Bz)2at the orbit. ForB&Vthgr;=0, the orbits are stable only ifns<1/2.