π Men hosts a transiting planet detected by the Transiting Exoplanet Survey Satellite space mission and an outer planet in a 5.7 yr orbit discovered by radial velocity (RV) surveys. We studied this system using new RV measurements taken with the HARPS spectrograph on ESO's 3.6 m telescope, as well as archival data. We constrain the stellar RV semiamplitude due to the transiting planet, π Men c, as K c = 1.21 ± 0.12 m s-1, resulting in a planet mass of M c = 3.63 ± 0.38 M ⊕. A planet radius of R c = 2.145 ± 0.015 R ⊕ yields a bulk density of ρ c = 2.03 ± 0.22 g cm-3. The precisely determined density of this planet and the brightness of the host star make π Men c an excellent laboratory for internal structure and atmospheric characterization studies. Our HARPS RV measurements also reveal compelling evidence for a third body, π Men d, with a minimum mass M d sin i d = 13.38 ± 1.35 M ⊕ orbiting with a period of P orb,d = 125 days on an eccentric orbit (e d = 0.22). A simple dynamical analysis indicates that the orbit of π Men d is stable on timescales of at least 20 Myr. Given the mutual inclination between the outer gaseous giant and the inner rocky planet and the presence of a third body at 125 days, π Men is an important planetary system for dynamical and formation studies.