  Pi Men hosts a transiting planet detected by the TESS space mission and an
outer planet in a 5.7-yr orbit discovered by RV surveys. We studied this system
using new radial velocity (RV) measurements taken with the HARPS spectrograph
on ESO's 3.6-m telescope as well as archival data. We constrain the stellar RV
semi-amplitude due to the transiting planet, Pi Men c, as K_c = 1.21 +/- 0.12
m/s resulting in a planet mass of M_c = 3.63 +/- 0.38 M_Earth. A planet radius
of R_c= 2.145 +/- 0.015 R_Earth yields a bulk density of rho = 2.03 +/- 0.22
g/cm^{-3}. The precisely determined density of this planet and the brightness
of the host star make Pi Men c an excellent laboratory for internal structure
and atmospheric characterization studies. Our HARPS RV measurements also reveal
compelling evidence for a third body, PI Men d, with a minimum mass M sin i =
13.38 +/- 1.35 M_Earth orbiting with a period of P_d = 125 d on an eccentric
orbit (e = 0.22). A simple dynamical analysis indicates that the orbit of Pi
Men d is stable on timescales of at least 20 Myrs. Given the mutual inclination
between the outer gaseous giant and the inner rocky planet and the presence of
a third body at 125 d, Pi Men is an important planetary system for dynamical
and formation studies.
