Context. Great strides have been made in recent years in the understanding of the mechanisms involved in the formation and evolution of planetary systems. Despite this, many observational findings have not yet been corroborated by astrophysical explanations. A fine contribution to the study of planetary formation processes comes from the study of young, low-mass planets, with short orbital periods (<100 days). In the last three years, the NASA/TESS satellite has identified many planets of this kind and their characterization is clearly necessary in order to understand how they formed and evolved. Aims. Within the framework of the Global Architecture of Planetary System (GAPS) project, we performed a validation and characterization (radius and mass) of the ultra-short period planet TOI-1807 b, which orbits its young host star BD+39 2643 (~300 Myr) in only 13 h. This is the youngest ultra-short period planet discovered so far. Methods. Thanks to a joint modeling of the stellar activity and planetary signals in the TESS light curve and in new HARPS-N radial-velocity measurements, combined with accurate estimation of stellar parameters, we validated the planetary nature of TOI-1807 b and measured its orbital and physical parameters. Results. By using astrometric, photometric, and spectroscopic observations, we found that BD+39 2643 is a young, active K dwarf star and a member of a 300 ± 80 Myr old moving group. Furthermore, it rotates in P_rot = 8.8 ± 0.1 days. This star hosts an ultra-short period planet, exhibiting an orbital period of only P_b = 0.54937 ± 0.00001 days. Thanks to the exquisite photometric and spectroscopic series, along with the accurate information on its stellar activity, we measured both the radius and the mass of TOI-1807 b with high precision, obtaining P_P,b = 1.37 ± 0.09 R_⊕ and M_P,b = 2.57 ± 0.50 M_⊕. These planet parameters correspond to a rocky planet with an Earth-like density (ρ_b = 1.0 ± 0.3 ρ_⊕) and no extended H/He envelope. From the analysis of the age-R_P distribution for planets with well measured ages, we inferred that TOI-1807 b may have already lost a large part of its atmosphere over the course of its 300 Myr lifetime.