(EDT-TTF-CONH2)6[Re6Se8(CN)6] is a molecular solid with R3̅ space group symmetry and has the remarkable feature of exhibiting hybrid donor layers with a kagome topology which sustain metallic conductivity. We report a detailed study of the structural evolution of the system as a function of temperature and pressure. This rhombohedral phase is maintained on cooling down to 220 K or up to 0.7 GPa pressure, beyond which a symmetry-breaking transition to a triclinic P1̅ phase drives a metal to insulator transition. Band structures calculated from the structural data lead to a clear description of the effects of temperature and pressure on the structural and electronic properties of this system. Linear energy dispersion is calculated at the zero-gap Fermi level where valence and conduction bands touch for the rhombohedral phase. (EDT-TTF-CONH2)6[Re6Se8(CN)6] thus exhibits a regular (right circular) Dirac-cone like that of graphene at the Fermi level, which has not been reported previously in a molecular solid. The Dirac-cone is robust over the stability region of the rhombohedral phase, and may result in exotic electronic transport and optical properties.