A 400 solar mass black hole in the Ultraluminous X-ray source M82 X-1 accreting close to its Eddington limit

M82 X-1, the brightest X-ray source in the galaxy M82, has been thought to be an intermediate-mass black hole (100 to 10,000 solar masses) because of its extremely high luminosity and variability characteristics, although some models suggest that its mass may be only about 20 solar masses. The previous mass estimates were based on scaling relations that use low-frequency characteristic timescales which have large intrinsic uncertainties. For stellar-mass black holes, we know that the high-frequency quasi-periodic oscillations (100-450 hertz) in the X-ray emission that occur in a 3:2 frequency ratio are stable and scale in frequency inversely with black hole mass with a reasonably small dispersion. The discovery of such stable oscillations thus potentially offers an alternative and less ambiguous means of mass determination for intermediate-mass black holes, but has hitherto not been realized. Here we report stable, twin-peak (3:2 frequency ratio) X-ray quasi-periodic oscillations from M82 X-1 at frequencies of 3.32$\pm$0.06 hertz and 5.07$\pm$0.06 hertz. Assuming that we can extrapolate the inverse-mass scaling that holds for stellar-mass black holes, we estimate the black hole mass of M82 X-1 to be 428$\pm$105 solar masses. In addition, we can estimate the mass using the relativistic precession model, from which we get a value of 415$\pm$63 solar masses.