We present experimental studies on ion acceleration from ultrathin diamondlike carbon foils irradiated by ultrahigh contrast laser pulses of energy 0.7 J focused to peak intensities of 5x10(19) W/cm2. A reduction in electron heating is observed when the laser polarization is changed from linear to circular, leading to a pronounced peak in the fully ionized carbon spectrum at the optimum foil thickness of 5.3 nm. Two-dimensional particle-in-cell simulations reveal that those C6+ ions are for the first time dominantly accelerated in a phase-stable way by the laser radiation pressure.