A few years ago H-6 Carbon was proposed as an all-sp2 three-dimensional carbon allotrope, with mechanical properties comparable to those of graphene. However, results on the stability of H-6 Carbon presented in the literature are rather contradictory and confusing, and it is not yet clear if this hypothetical allotrope is stable or not. Studying systematically the stability of H-6 Carbon, using ab initio density functional theory and phonon band structure calculations, we show that H-6 Carbon is unstable, converted spontaneously to diamond. According to our findings, the instability mechanism is not the same as that of compressed rhombohedral graphite, but is related to the synergetic action of the interchain interactions of the parallel-arranged zig-zag chains and the strain induced by the 60[degree] rotation (with respect to graphite) of the interconnected zig-zag chains. This synergetic action eliminates the barrier provided by the intrachain interactions, causing the transition of H-6 Carbon to diamond.