We present experimental studies on ion acceleration from ultra-thin diamond-like carbon (DLC) foils irradiated by ultra-high contrast laser pulses of energy 0.7 J focussed to peak intensities of 5*10^{19} W/cm^2. 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 (PIC) simulations reveal, that those C^{6+} ions are for the first time dominantly accelerated in a phase-stable way by the laser radiation pressure.
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