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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., 15 pages, 4 figures. Accepted for publication in APJL

HATSouth is the world's first network of automated and homogeneous telescopes that is capable of year-round 24-hour monitoring of positions over an entire hemisphere of the sky. The primary scientific goal of the network is to discover and characterize a large number of transiting extrasolar planets, reaching out to long periods and down to small planetary radii. HATSouth achieves this by monitoring extended areas on the sky, deriving high precision light curves for a large number of stars, searching for the signature of planetary transits, and confirming planetary candidates with larger telescopes. HATSouth employs 6 telescope units spread over 3 locations with large longitude separation in the southern hemisphere (Las Campanas Observatory, Chile; HESS site, Namibia; Siding Spring Observatory, Australia). Each of the HATSouth units holds four 0.18m diameter f/2.8 focal ratio telescope tubes on a common mount producing an 8.2x8.2 arcdeg field, imaged using four 4Kx4K CCD cameras and Sloan r filters, to give a pixel scale of 3.7 arcsec/pixel. The HATSouth network is capable of continuously monitoring 128 square arc-degrees. We present the technical details of the network, summarize operations, and present weather statistics for the 3 sites. On average each of the 6 HATSouth units has conducted observations on ~500 nights over a 2-year time period, yielding a total of more than 1million science frames at 4 minute integration time, and observing ~10.65 hours per day on average. We describe the scheme of our data transfer and reduction from raw pixel images to trend-filtered light curves and transiting planet candidates. Photometric precision reaches ~6 mmag at 4-minute cadence for the brightest non-saturated stars at r~10.5. We present detailed transit recovery simulations to determine the expected yield of transiting planets from HATSouth. (abridged), 25 pages, 11 figures, 1 table, submitted to PASP