  We report the detection of GJ 3090 b (TOI-177.01), a mini-Neptune on a
2.9-day orbit transiting a bright (K = 7.3 mag) M2 dwarf located at 22 pc. The
planet was identified by the Transiting Exoplanet Survey Satellite and was
confirmed with the High Accuracy Radial velocity Planet Searcher radial
velocities. Seeing-limited photometry and speckle imaging rule out nearby
eclipsing binaries. Additional transits were observed with the LCOGT, Spitzer,
and ExTrA telescopes. We characterise the star to have a mass of 0.519 $\pm$
0.013 M$_\odot$ and a radius of 0.516 $\pm$ 0.016 R$_\odot$. We modelled the
transit light curves and radial velocity measurements and obtained a planetary
mass of 3.34 $\pm$ 0.72 M$_\oplus$, a radius of 2.13 $\pm$ 0.11 R$_\oplus$, and
a mean density of 1.89$^{+0.52}_{-0.45}$ g/cm$^3$. The low density of the
planet implies the presence of volatiles, and its radius and insolation place
it immediately above the radius valley at the lower end of the mini-Neptune
cluster. A coupled atmospheric and dynamical evolution analysis of the planet
is inconsistent with a pure H-He atmosphere and favours a heavy mean molecular
weight atmosphere. The transmission spectroscopy metric of 221$^{+66}_{-46}$
means that GJ 3090 b is the second or third most favourable mini-Neptune after
GJ 1214 b whose atmosphere may be characterised. At almost half the mass of GJ
1214 b, GJ 3090 b is an excellent probe of the edge of the transition between
super-Earths and mini-Neptunes. We identify an additional signal in the radial
velocity data that we attribute to a planet candidate with an orbital period of
13 days and a mass of 17.1$^{+8.9}_{-3.2}$ M$_\oplus$, whose transits are not
detected.
