α-Graphyne, a carbon-expanded version of graphene ('carbo-graphene') that was recently evidenced as an alternative zero-gap semiconductor, remains a theoretical material. Nevertheless, using specific synthesis methods, molecular units of α-graphyne ('carbo-benzene' macrocycles) can be inserted between two anilinyl (4-NH2-C6H4)-anchoring groups that allow these fragments to form molecular junctions between gold electrodes. Here, electrical measurements by the scanning tunnelling microscopy (STM) break junction technique and electron transport calculations are carried out on such a carbo-benzene, providing unprecedented single molecule conductance values: 106 nS through a 1.94-nm N-N distance, essentially 10 times the conductance of a shorter nanographenic hexabenzocoronene analogue. Deleting a C4 edge of the rigid C18 carbo-benzene circuit results in a flexible 'carbo-butadiene' molecule that has a conductance 40 times lower. Furthermore, carbo-benzene junctions exhibit field-effect transistor behaviour when an electrochemical gate potential is applied, opening the way for device applications. All the results are interpreted on the basis of theoretical calculations.