We report the detection of a transiting planet around $pi$ Mensae (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 $2.04pm 0.05$ $R_oplus$ 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 $4.82pm 0.85$ $M_oplus$. The stars proximity and brightness will facilitate further investigations, such as atmospheric spectroscopy, asteroseismology, the Rossiter--McLaughlin effect, astrometry, and direct imaging.
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