Thin films of molecular organic semiconductors are attracting much interest for use in electronic and optoelectronic applications. The electronic properties of these materials and their interfaces are therefore worth investigating intensively, particularly the degree of electron delocalization that can be achieved. If the delocalization is appreciable, it should be accompanied by an observable electronic band dispersion. But so far only limited experimental data on the intermolecular dispersion of electronic states in molecular materials is available, and the mechanism(s) of electron delocalization in molecular materials are also not well understood. Here we report scanning tunnelling spectroscopy observations of an organic monolayer film on a silver substrate, revealing a completely delocalized two-dimensional band state that is characterized by a metal-like parabolic dispersion with an effective mass of m* = 0.47m(e), where m(e) is the bare electron mass. This dispersion is far stronger than expected for the organic film alone, and arises as a result of strong substrate-mediated coupling between the molecules within the monolayer.