  We present the discovery of TOI-3235 b, a short-period Jupiter orbiting an
M-dwarf with a stellar mass close to the critical mass at which stars
transition from partially to fully convective. TOI-3235 b was first identified
as a candidate from TESS photometry, and confirmed with radial velocities from
ESPRESSO, and ground-based photometry from HATSouth, MEarth-South,
TRAPPIST-South, LCOGT, and ExTrA. We find that the planet has a mass of
$\mathrm{0.665\pm0.025\,M_J}$ and a radius of $\mathrm{1.017\pm0.044\,R_J}$. It
orbits close to its host star, with an orbital period of $\mathrm{2.5926\,d}$,
but has an equilibrium temperature of $\mathrm{\approx 604 \, K}$, well below
the expected threshold for radius inflation of hot Jupiters. The host star has
a mass of $\mathrm{0.3939\pm0.0030\,M_\odot}$, a radius of
$\mathrm{0.3697\pm0.0018\,R_\odot}$, an effective temperature of $\mathrm{3389
\, K}$, and a J-band magnitude of $\mathrm{11.706\pm0.025}$. Current planet
formation models do not predict the existence of gas giants such as TOI-3235 b
around such low-mass stars. With a high transmission spectroscopy metric,
TOI-3235 b is one of the best-suited giants orbiting M-dwarfs for atmospheric
characterization.
