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 0.665+/-0.025M_J_ and a radius of 1.017+/-0.044R_J_. It orbits close to its host star, with an orbital period of 2.5926 days but has an equilibrium temperature of ~604K, well below the expected threshold for radius inflation of hot Jupiters. The host star has a mass of 0.3939+/-0.0030M_{sun}, a radius of 0.3697+/-0.0018R{sun}_, an effective temperature of 3389K, and a J-band magnitude of 11.706+/-0.025. 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.