Growing numbers of exoplanet detections continue to reveal the diverse nature of planetary systems. Planet formation around late-type M dwarfs is of particular interest. These systems provide practical laboratories to measure exoplanet occurrence rates for M dwarfs, thus testing how the outcomes of planet formation scale with host mass, and how they compare to Sun-like stars. Here, we report the discovery of TOI-6478 b, a cold ($T_{\text{eq}}=204\,$ K) Neptune-like planet orbiting an M5 star ($R_\star =0.234\pm 0.012\, \text{R}_\odot$, $M_\star =0.230\pm 0.007\, \text{M}_\odot$, $T_{\text{eff}}=3230\pm 75\,$ K) that is a member of the Milky Way's thick disc. We measure a planet radius of $R_b=4.6\pm 0.24\, \text{R}_{\oplus }$ on a $P_b=34.005019\pm 0.000025\,$ d orbit. Using radial velocities, we calculate an upper mass limit of $M_b\le 9.9\, \text{M}_{\oplus }$ ($M_b\le 0.6\, \text{M}_{\text{Nep}})$, with $3\, \sigma$ confidence. TOI-6478 b is a milestone planet in the study of cold Neptune-like worlds. Due to its large atmospheric scale height, it is amenable to atmospheric characterization with facilities such as JWST, and will provide an excellent probe of atmospheric chemistry in this cold regime. It is one of very few transiting exoplanets that orbit beyond their system's ice-line whose atmospheric chemical composition can be measured. Based on our current understanding of this planet, we estimate TOI-6478 b's spectroscopic features (in transmission) can be $\sim 2.5\times$ as high as the widely studied planet K2-18 b.