Context. Although more than 2000 brown dwarfs have been detected to date, mainly from direct imaging, their characterisation is difficult due to their faintness and model-dependent results. In the case of transiting brown dwarfs, however, it is possible to make direct high-precision observations. Aims: Our aim is to investigate the nature and formation of brown dwarfs by adding a new well-characterised object, in terms of its mass, radius and bulk density, to the currently small sample of less than 20 transiting brown dwarfs. Methods: One brown dwarf candidate was found by the KESPRINT consortium when searching for exoplanets in the K2 space mission Campaign 16 field. We combined the K2 photometric data with a series of multicolour photometric observations, imaging, and radial velocity measurements to rule out false positive scenarios and to determine the fundamental properties of the system. Results: We report the discovery and characterisation of a transiting brown dwarf in a 5.17-day eccentric orbit around the slightly evolved F7 V star EPIC 212036875. We find a stellar mass of 1.15 ± 0.08 M⊙, a stellar radius of 1.41 ± 0.05 R⊙, and an age of 5.1 ± 0.9 Gyr. The mass and radius of the companion brown dwarf are 51 ± 2 MJ and 0.83 ± 0.03 RJ, respectively, corresponding to a mean density of 108-13+15 g cm-3. Conclusions: EPIC 212036875 b is a rare object that resides in the brown-dwarf desert. In the mass-density diagram for planets, brown dwarfs, and stars, we find that all giant planets and brown dwarfs follow the same trend from 0.3 MJ to the turn-over to hydrogen burning stars at 73 MJ. EPIC 212036875 b falls close to the theoretical model for mature H/He dominated objects in this diagram as determined by interior structure models. We argue that EPIC 212036875 b formed via gravitational disc instabilities in the outer part of the disc, followed by a quick migration. Orbital tidal circularisation may have started early in its history for a brief period when the brown dwarf's radius was larger. The lack of spin-orbit synchronisation points to a weak stellar dissipation parameter (Q⋆' ≳ 108), which implies a circularisation timescale of ≳23 Gyr, or suggests an interaction between the magnetic and tidal forces of the star and the brown dwarf. Table 2 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/628/A64This work is done under the framework of the KESPRINT collaboration (http://kesprint.science). KESPRINT is an international consortium devoted to the characterisation and research of exoplanets discovered with space-based missions.