We report the discovery of a super-Earth orbiting at the inner edge of the habitable zone of the star GJ 625 based on the analysis of the radial-velocity (RV) time series from the HARPS-N spectrograph, consisting in 151 HARPS-N measurements taken over 3.5 yr. GJ 625 b is a planet with a minimum mass M sin $i$ of 2.82 $\pm$ 0.51 M$_{\oplus}$ with an orbital period of 14.628 $\pm$ 0.013 days at a distance of 0.078 AU of its parent star. The host star is the quiet M2 V star GJ 625, located at 6.5 pc from the Sun. We find the presence of a second radial velocity signal in the range 74-85 days that we relate to stellar rotation after analysing the time series of Ca II H\&K and H${\alpha}$ spectroscopic indicators, the variations of the FWHM of the CCF and and the APT2 photometric light curves. We find no evidence linking the short period radial velocity signal to any activity proxy.