  The Transiting Exoplanet Survey Satellite, TESS, is currently carrying out an
all-sky search for small planets transiting bright stars. In the first year of
the TESS survey, steady progress was made in achieving the mission's primary
science goal of establishing bulk densities for 50 planets smaller than
Neptune. During that year, TESS's observations were focused on the southern
ecliptic hemisphere, resulting in the discovery of three mini-Neptunes orbiting
the star TOI-125, a V=11.0 K0 dwarf. We present intensive HARPS radial velocity
observations, yielding precise mass measurements for TOI-125b, TOI-125c and
TOI-125d. TOI-125b has an orbital period of 4.65 days, a radius of $2.726 \pm
0.075 ~\mathrm{R_{\rm E}}$, a mass of $ 9.50 \pm 0.88 ~\mathrm{M_{\rm E}}$ and
is near the 2:1 mean motion resonance with TOI-125c at 9.15 days. TOI-125c has
a similar radius of $2.759 \pm 0.10 ~\mathrm{R_{\rm E}}$ and a mass of $ 6.63
\pm 0.99 ~\mathrm{M_{\rm E}}$, being the puffiest of the three planets.
TOI-125d, has an orbital period of 19.98 days and a radius of $2.93 \pm
0.17~\mathrm{R_{\rm E}}$ and mass $13.6 \pm 1.2 ~\mathrm{M_{\rm E}}$. For
TOI-125b and TOI-125d we find unusual high eccentricities of $0.19\pm 0.04$ and
$0.17^{+0.08}_{-0.06}$, respectively. Our analysis also provides upper mass
limits for the two low-SNR planet candidates in the system; for TOI-125.04
($R_P=1.36 ~\mathrm{R_{\rm E}}$, $P=$0.53 days) we find a $2\sigma$ upper mass
limit of $1.6~\mathrm{M_{\rm E}}$, whereas TOI-125.05 ( $R_P=4.2^{+2.4}_{-1.4}
~\mathrm{R_{\rm E}}$, $P=$ 13.28 days) is unlikely a viable planet candidate
with upper mass limit $2.7~\mathrm{M_{\rm E}}$. We discuss the internal
structure of the three confirmed planets, as well as dynamical stability and
system architecture for this intriguing exoplanet system.
