We report the detection of a transiting planet around π Men (HD 39091), using data from the Transiting Exoplanet Survey Satellite (TESS). The solar-type host star is unusually bright (V = 5.7) and was already known to host a Jovian planet on a highly eccentric, 5.7-year orbit. The newly discovered planet has a size of 2.04 ± 0.05 R⊕ and an orbital period of 6.27 days. Radial-velocity data from the HARPS and AAT/UCLES archives also displays a 6.27-day periodicity, confirming the existence of the planet and leading to a mass determination of 4.82±0.85 M⊕. The star's proximity and brightness will facilitate further investigations, such as atmospheric spectroscopy, asteroseismology, the Rossiter-McLaughlin effect, astrometry, and direct imaging.