We report generation of a femtosecond supercontinuum extending from the ultraviolet to the near-infrared and detection of its carrier-envelope phase variation by f-to-2f interferometry. The spectrum is generated in a gas-filled hollow-core photonic crystal fiber where soliton dynamics allows CEP-stable self-compression of OPCPA pump pulses at 800 nm to a duration of 1.7 optical cycles, followed by dispersive wave emission. The source provides up to 1 μJ of pulse energy at 800 kHz repetition rate resulting in 0.8 W of average power, and can be extremely useful for example in strong-field physics, pump-probe measurements and ultraviolet frequency comb metrology.