Planets with orbital periods shorter than 1 day are rare and have formation histories that are not completely understood. Small ($R_\mathrm{p} < 2\; R_\oplus$) ultra-short-period (USP) planets are highly irradiated, probably have rocky compositions with high bulk densities, and are often found in multi-planet systems. Additionally, USP planets found around small stars are excellent candidates for characterization using present-day instrumentation. Of the current full sample of approximately 5500 confirmed exoplanets, only 130 are USP planets and around 40 have mass and radius measurements. Wolf 327 (TOI-5747) is an M dwarf ($R_\star = 0.406 \pm 0.015 \; R_\odot$, $M_\star = 0.405 \pm 0.019 \; M_\odot$, $T_{\mathrm{eff}}=3542 \pm 70$ K, and $V = 13$ mag) located at a distance $d = 28.5$ pc. NASA's planet hunter satellite, TESS, detected transits in this star with a period of 0.573 d (13.7 h) and with a transit depth of 818 ppm. Ground-based follow-up photometry, high resolution imaging, and radial velocity (RV) measurements taken with the CARMENES spectrograph confirm the presence of this new USP planet. Wolf 327b is a super-Earth with a radius of $R_\mathrm{p} = 1.24 \pm 0.06 \; R_\oplus$ and a mass of $M_\mathrm{p} = 2.53 \pm 0.46 \; M_\oplus$, yielding a bulk density of $7.24 \pm 1.66 $ g cm$^{-3}$ and thus suggesting a rocky composition. Owing to its close proximity to its host star ($a = 0.01$ au), Wolf 327b has an equilibrium temperature of $996 \pm 22$ K. This planet has a mass and radius similar to K2-229b, a planet with an inferred Mercury-like internal composition. Planet interior models suggest that Wolf 327b has a large iron core, a small rocky mantle, and a negligible (if any) H/He atmosphere.