The functionalities offered by single-molecule electrical junctions are yet to be translated into monolayer or few-layer molecular films, where making effective and reproducible electrical contact is one of the challenging bottlenecks. Here we take a significant step in this direction by demonstrating that excellent electrical contact can be made with a monolayer biphenyl-4,4′-dithiol (BPDT) molecular film, sandwiched between gold and graphene electrodes. This sandwich device structure is advantageous, because the current flows through the molecules to the gold substrate in a ‘cross-plane’ manner, perpendicular to the plane of graphene, yielding high-conductance devices. We elucidate the nature of the cross-plane graphene/molecule/Au transport using quantum transport calculations and introduce a simple analytical model, which captures generic features of the current–voltage characteristic. Asymmetry in junction properties results from the disparity in electrode electrical properties, the alignment of the BPDT HOMO–LUMO energy levels and the specific characteristics of the graphene electrode. The experimental observation of scalability of junction properties within the junction area, in combination with a theoretical description of the transmission probability of the thiol–graphene contact, demonstrates that between 10% and 100% of the molecules make contact with the electrodes, which is several orders of magnitude greater than that achieved to date in the literature.