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