A stable relativistic ion acceleration regime for thin foils irradiated by circularly polarized laser pulses is suggested. In this regime, the "light-sail" stage of radiation pressure acceleration for ions is smoothly connected with the initial relativistic "hole-boring" stage, and a defined relationship between laser intensity I0, foil density n{0}, and thickness l{0} should be satisfied. For foils with a wide range of n{0}, the required I0 and l{0} for the regime are theoretically estimated and verified with the particle-in-cell code ILLUMINATION. It is shown for the first time by 2D simulations that high-density monoenergetic ion beams with energy above GeV/u and divergence of 10 degrees are produced by circularly polarized lasers at intensities of 10;{22} W/cm;{2}, which are within reach of current laser systems.