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.