We present X-ray/γ-ray spectra of Cyg X-1 observed during the transition from the hard to the soft state and in the soft state by ASCA, RXTE and CGRO/OSSE in 1996 May and June. The spectra consist of a dominant soft component below ~2keV and a power-law-like continuum extending to at least ~800keV. We interpret them as emission from an optically thick, cold accretion disc and from an optically thin, non-thermal corona above the disc. A fraction f>~0.5 of total available power is dissipated in the corona. We model the soft component by multicolour blackbody disc emission taking into account the torque-free inner-boundary condition. If the disc extends down to the minimum stable orbit, the ASCARXTE data yield the most probable black hole mass of MX~10Msolar and an accretion rate, Msolar~0.5LEc2, locating Cyg X-1 in the soft state in the upper part of the stable, gas-pressure-dominated, accretion-disc solution branch. The spectrum of the corona is well modelled by repeated Compton scattering of seed photons from the disc off electrons with a hybrid, thermal/non-thermal distribution. The electron distribution can be characterized by a Maxwellian with an equilibrium temperature of kTe~30-50keV, a Thomson optical depth of τ~0.3 and a quasi-power-law tail. The compactness of the corona is 2<~lh<~7, and a presence of a significant population of electron-positron pairs is ruled out. We find strong signatures of Compton reflection from a cold and ionized medium, presumably an accretion disc, with an apparent reflector solid angle, Ω/2π~0.5-0.7. The reflected continuum is accompanied by a broad iron Kα line.