About one out of 200 Sun-like stars has a planet with an orbital period shorter than one day: an ultra-short-period planet (Sanchis-ojeda et al. 2014; Winn et al. 2018). All of the previously known ultra-short-period planets are either hot Jupiters, with sizes above 10 Earth radii (Re), or apparently rocky planets smaller than 2 Re. Such lack of planets of intermediate size (the "hot Neptune desert") has been interpreted as the inability of low-mass planets to retain any hydrogen/helium (H/He) envelope in the face of strong stellar irradiation. Here, we report the discovery of an ultra-short-period planet with a radius of 4.6 Re and a mass of 29 Me, firmly in the hot Neptune desert. Data from the Transiting Exoplanet Survey Satellite (Ricker et al. 2015) revealed transits of the bright Sun-like star \starname\, every 0.79 days. The planet's mean density is similar to that of Neptune, and according to thermal evolution models, it has a H/He-rich envelope constituting 9.0^(+2.7)_(-2.9)% of the total mass. With an equilibrium temperature around 2000 K, it is unclear how this "ultra-hot Neptune" managed to retain such an envelope. Follow-up observations of the planet's atmosphere to better understand its origin and physical nature will be facilitated by the star's brightness (Vmag=9.8).