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Aims. We report the confirmation of a new transiting exoplanet orbiting the star TOI-5076. Methods. We present our vetting procedure and follow-up observations which led to the confirmation of the exoplanet TOI-5076b. In particular, we employed high-precision {\it TESS} photometry, high-angular-resolution imaging from several telescopes, and high-precision radial velocities from HARPS-N. Results. From the HARPS-N spectroscopy, we determined the spectroscopic parameters of the host star: T$\rm_{eff}$=(5070$\pm$143) K, log~g=(4.6$\pm$0.3), [Fe/H]=(+0.20$\pm$0.08), and [$\alpha$/Fe]=0.05$\pm$0.06. The transiting planet is a warm sub-Neptune with a mass m$\rm_p=$(16$\pm$2) M$\rm_{\oplus}$, a radius r$\rm_p=$(3.2$\pm$0.1)~R$\rm_{\oplus}$ yielding a density $\rho_p$=(2.8$\pm$0.5) g cm$^{-3}$. It revolves around its star approximately every 23.445 days. Conclusions. The host star is a metal-rich, K2V dwarf, located at about 82 pc from the Sun with a radius of R$_{\star}$=(0.78$\pm$0.01) R$_{\odot}$ and a mass of M$_{\star}$=(0.80$\pm$0.07) M$_{\odot}$. It forms a common proper motion pair with an M-dwarf companion star located at a projected separation of 2178 au. The chemical analysis of the host-star and the Galactic-space velocities indicate that TOI-5076 belongs to the old population of thin-to-thick-disk transition stars. The density of TOI-5076b suggests the presence of a large fraction by volume of volatiles overlying a massive core. We found that a circular orbit solution is marginally favored with respect to an eccentric orbit solution for TOI-5076b., Comment: Accepted by Astronomy & Astrophysics: 15 pages, 19 figures, 4 tables