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. © 2018 The Royal Society of Chemistry.