Most of our knowledge of extrasolar planets rests on precise radial-velocity measurements, either for direct detection or for confirmation of the planetary origin of photometric transit signals. This has limited our exploration of the parameter space of exoplanet hosts to solar- and later-type, sharp-lined stars. Here we extend the realm of stars with known planetary companions to include hot, fast-rotating stars. Planet-like transits have previously been reported in the light curve obtained by the SuperWASP survey of the A5 star HD 15082 (WASP–33; V= 8.3, v sin i= 86 km s−1). Here we report further photometry and time-series spectroscopy through three separate transits, which we use to confirm the existence of a gas-giant planet with an orbital period of 1.22 d in orbit around HD 15082. From the photometry and the properties of the planet signal travelling through the spectral line profiles during the transit, we directly derive the size of the planet, the inclination and obliquity of its orbital plane and its retrograde orbital motion relative to the spin of the star. This kind of analysis opens the way to studying the formation of planets around a whole new class of young, early-type stars, hence under different physical conditions and generally in an earlier stage of formation than in sharp-lined late-type stars. The reflex orbital motion of the star caused by the transiting planet is small, yielding an upper mass limit of 4.1 MJupiter on the planet. We also find evidence of a third body of substellar mass in the system, which may explain the unusual orbit of the transiting planet. In HD 15082, the stellar line profiles also show evidence of non-radial pulsations, clearly distinct from the planetary transit signal. This raises the intriguing possibility that tides raised by the close-in planet may excite or amplify the pulsations in such stars.