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 ∼3.49 days. Joint fitting of the radial velocities and TESS and ground-based transits yields a planetary radius of 5.05-0.17+0.16 R⊕, mass 26.78-5.13+5.43 M⊕, and eccentricity 0.057-0.039+0.054. We estimate the stellar properties, and perform a search for He 10830 Å 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 Å absorption has been detected in the planet atmosphere of GJ 3470b, 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.