Magnetoresistance is a change in the resistance of a material system caused by an applied magnetic field. Giant magnetoresistance occurs in structures containing ferromagnetic contacts separated by a metallic non-magnetic spacer, and is now the basis of read heads for hard drives and for new forms of random access memory. Using an insulator (for example, a molecular thin film) rather than a metal as the spacer gives rise to tunnelling magnetoresistance, which typically produces a larger change in resistance for a given magnetic field strength, but also yields higher resistances, which are a disadvantage for real device operation. Here, we demonstrate giant magnetoresistance across a single, non-magnetic hydrogen phthalocyanine molecule contacted by the ferromagnetic tip of a scanning tunnelling microscope. We measure the magnetoresistance to be 60% and the conductance to be 0.26G(0), where G(0) is the quantum of conductance. Theoretical analysis identifies spin-dependent hybridization of molecular and electrode orbitals as the cause of the large magnetoresistance.