Context. One of the goals of the ESPRESSO guaranteed time observations (GTOs) at the ESO 8.2 m telescope is to follow up on candidate planets from transit surveys such as the TESS mission. High-precision radial velocities are required to characterize small exoplanets. Aims. We intend to confirm the existence of a transiting super-Earth around the bright (V=9.74) K0-type star TOI-512 (TIC 119292328) and provide a characterization. Methods. Combining photometric data from TESS and 37 high-resolution spectroscopic observations from ESPRESSO in a joint Markov chain Monte Carlo analysis, we determined the planetary parameters of TOI-512b and characterized its internal structure. Results. We find that TOI-512b is a super-Earth, with a radius of 1.54 ± 0.10 R⊕ and mass of 3.57‑0.55+0.53 M⊕, on a 7.19‑6.1·10‑5+7.10‑5 day orbit. This corresponds to a bulk density of 5.62‑1.28+1.59 g cm‑3. Our interior structure analysis presents a small inner core representing 0.13‑0.11+0.13 of the solid mass fraction for the planet, surrounded by a mantle with a mass fraction of 0.69‑0.22+0.20, and an upper limit of the water layer of 0.16. The gas mass below 10‑8.93 indicates a very small amount of gas on the planet. We find no evidence of the second candidate found by the TESS pipeline, TOI-512.02, neither in TESS photometry, nor in the ESPRESSO radial velocities. The low stellar activity makes it an interesting transmission spectroscopy candidate for future-generation instruments. ★Based on Guaranteed Time Observations collected at the European Southern Observatory by the ESPRESSO Consortium under ESO programmes 106.21M2.007, 108.2254.002, 108.2254.005, and 108.2254.006.