We report on the determination of the mass of TOI-519 b, a transiting substellar object around a mid-M dwarf. We carried out radial velocity measurements using Subaru/InfraRed Doppler (IRD), revealing that TOI-519 b is a planet with a mass of $0.463^{+0.082}_{-0.088}\, M_{\rm Jup}$. We also found that the host star is metal rich ([Fe/H] = 0.27 ± 0.09 dex) and has the lowest effective temperature (Teff = 3322 ± 49 K) among all stars hosting known close-in giant planets based on the IRD spectra and mid-resolution infrared spectra obtained with NASA Infrared Telescope Facility/SpeX. The core mass of TOI-519 b inferred from a thermal evolution model ranges from 0 to ~30 M⊕, which can be explained by both core accretion and disk instability models as the formation origins of this planet. However, TOI-519 is in line with the emerging trend that M dwarfs with close-in giant planets tend to have high metallicity, which may indicate that they formed in the core accretion model. The system is also consistent with the potential trend that close-in giant planets around M dwarfs tend to be less massive than those around FGK dwarfs.