A new ultra-hard rhombohedral carbon rh-C4 (or hexagonal h-C12) is reported as derived from 3R graphite through crystal chemistry construction and ground state energy within the density functional theory. A resulting cohesive extended hexagonal three-dimensional network of h-C12 is identified as formed of C4 tetrahedra like in cubic and hexagonal diamond. The electronic band structure of rh-C4 is characteristic of insulator with Egap ​= ​4 ​eV similarly to diamond. From the energy-volume equations of state and the set of elastic constants a larger value of bulk modulus versus lonsdaleite, and the largest Vickers hardness (Hv) versus both forms of diamond were revealed. The phonon band structure showing only real wave numbers further confirms the stability of the new carbon phase.