In this paper we report the discovery of TOI-220 $b$, a new sub-Neptune detected by the Transiting Exoplanet Survey Satellite (TESS) and confirmed by radial velocity follow-up observations with the HARPS spectrograph. Based on the combined analysis of TESS transit photometry and high precision radial velocity measurements we estimate a planetary mass of 13.8 $pm$ 1.0 M$_{Earth}$ and radius of 3.03 $pm$ 0.15 R$_{Earth}$, implying a bulk density of 2.73 $pm$ 0.47 $textrm{g cm}^{-3}$. TOI-220 $b$ orbits a relative bright (V=10.4) and old (10.1$pm$1.4 Gyr) K dwarf star with a period of $sim$10.69 d. Thus, TOI-220 $b$ is a new warm sub-Neptune with very precise mass and radius determinations. A Bayesian analysis of the TOI-220 $b$ internal structure indicates that due to the strong irradiation it receives, the low density of this planet could be explained with a steam atmosphere in radiative-convective equilibrium and a supercritical water layer on top of a differentiated interior made of a silicate mantle and a small iron core.
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