As early as 10 Gyr ago, galaxies with more than 10^11 M_sun in stars already existed. While most of these  massive galaxies must have subsequently transformed through on-going star formation and mergers with other galaxies, a small fraction (<0.1%) may have survived untouched till today. Searches for such relic galaxies, useful windows to explore the early Universe, have been inconclusive to date: galaxies with masses and sizes like those observed at high redshift (M_*>10^11 M_sun; R_e<1.5 kpc) have been found in the local Universe, but their stars are far too young for the galaxy to be a relic galaxy. This paper explores the first case of a nearby galaxy, NGC1277 (at a distance of 73 Mpc in the Perseus galaxy cluster), which fulfils many criteria to be considered a relic galaxy. Using deep optical spectroscopy, we derive the star formation history along the structure of the galaxy: the stellar populations are uniformly old (> 10 Gyr) with no evidence for more recent star formation episodes. The metallicity of their stars is super-solar ([Fe/H]=0.20+-0.04 with a smooth decline towards the outer regions) and alpha enriched ([alpha/Fe]=0.4+-0.1). This suggests a very short formation time scale for the bulk of stars of this galaxy. This object also rotates very fast (V_rot~300 km/s) and has a large central velocity dispersion (sigma>300 km/s).  NGC1277  allows the explorations in full detail of properties such as the structure, internal dynamics, metallicity and initial mass function at ~10-12 Gyr back in time when the first massive galaxies were built.