Small-sized ($<4$ $R_ oplus $) exoplanets in tight orbits around young stars (10--1000 Myr) give us the opportunity to investigate the mechanisms that led to their formation, the evolution of their physical and orbital properties, and, in particular, their atmospheres. Thanks to the all-sky survey carried out by the spacecraft, many of these exoplanets have been discovered, and have subsequently been characterized with dedicated follow-up observations. In the context of a collaboration among the Global Architecture of Planetary Systems (GAPS) team, the TESS-Keck Survey (TKS) team, and the California Planet Search (CPS) team, we measured ---with a high level of precision--- the mass and the radius of TOI-1430 b, a young ($ 700$ Myr) exoplanet with an escaping He atmosphere orbiting the K-dwarf star HD 235088 (TOI-1430). By adopting appropriate stellar parameters, which were measured in this work, we were able to simultaneously model the signals due to strong stellar activity and the transiting planet TOI-1430 b in both photometric and spectroscopic series. This allowed us to measure both the radius and mass (and consequently the density) of the planet with high precision, and to reconstruct the evolution of its atmosphere. TOI-1430 is an active K-dwarf star born $700 150$ Myr ago, with a rotation period of rot 12$ days. This star hosts a mini-Neptune, whose orbital period is b 0.000004$ days. Thanks to long-term photometric and spectroscopic monitoring of this target performed with TESS, HARPS-N, HIRES, and APF, we estimate a radius of P,b oplus $, a mass of $M_ P,b oplus $, and thus a planetary density of $ b oplus $. TOI-1430 b is therefore a low-density mini-Neptune with an extended atmosphere, and is at the edge of the radius gap. Because this planet is known to have an evaporating atmosphere of He, we reconstructed its atmospheric history. Our analysis supports the scenario in which, shortly after its birth, TOI-1430 b was super-puffy, with a radius $5 and a mass $1.5 the values of today; in $ 200$ Myr from now, TOI-1430 b should lose its envelope, showing its Earth-size core. We also looked for signals from a second planet in the spectroscopic and photometric series, without detecting any.