We confirm the planetary nature of TOI-1728b using a combination of ground-based photometry, near-infrared Doppler velocimetry and spectroscopy with the Habitable-zone Planet Finder. TOI-1728 is an old, inactive M0 star with Teff=3980_-32_^+31^K, which hosts a transiting super-Neptune at an orbital period of ~3.49days. Joint fitting of the radial velocities and TESS and ground-based transits yields a planetary radius of 5.05_-0.17_^+0.16^ R{Earth}, mass 26.78_-5.13_^+5.43^M{Earth}, and eccentricity 0.057_-0.039_^+0.054^. We estimate the stellar properties, and perform a search for He 10830{AA} absorption during the transit of this planet and claim a null detection with an upper limit of 1.1% with 90% confidence. A deeper level of He 10830{AA} absorption has been detected in the planet atmosphere of GJ3470b, a comparable gaseous planet. TOI-1728b is the largest super-Neptune-the intermediate subclass of planets between Neptune and the more massive gas-giant planets-discovered around an M-dwarf. With its relatively large mass and radius, TOI-1728 represents a valuable data point in the M-dwarf exoplanet mass-radius diagram, bridging the gap between the lighter Neptune-sized planets and the heavier Jovian planets known to orbit M dwarfs. With a low bulk density of 1.14_-0.24_^+0.26^g/cm^3^, and orbiting a bright host star (J~9.6, V~12.4), TOI-1728b is also a promising candidate for transmission spectroscopy both from the ground and from space, which can be used to constrain planet formation and evolutionary models.