We report the discovery of an extremely close white dwarf plus F dwarf main-sequence star in a 12 h binary identified by combining data from the Radial Velocity Experiment survey and the Galaxy Evolution Explorer survey. A combination of spectral energy distribution fitting and optical and Hubble Space Telescope ultraviolet spectroscopy allowed us to place fairly precise constraints on the physical parameters of the binary. The system, TYC 6760-497-1, consists of a hot Teff ~ 20 000 K, MWD ~ 0.6M⊙ white dwarf and an F8 star (MMS ~ 1.23M⊙, RMS ~ 1.3R⊙) seen at a low inclination (i ~ 37°). The system is likely the descendant of a binary that contained the F star and an ~2M⊙ A-type star that filled its Roche lobe on the thermally pulsating asymptotic giant branch, initiating a common envelope phase. The F star is extremely close to Roche lobe filling and there is likely to be a short phase of thermal time-scale mass transfer on to the white dwarf during which stable hydrogen burning occurs. During this phase, it will grow in mass by up to 20 per cent, until the mass ratio reaches close to unity, at which point it will appear as a standard cataclysmic variable star. Therefore, TYC 6760-497-1 is the first known progenitor of a supersoft source system, but will not undergo a Type Ia supernova explosion. Once an accurate distance to the system is determined by Gaia, we will be able to place very tight constraints on the stellar and binary parameters.