Metal nanostructures act as powerful optical antennas(1,2) because collective modes of the electron fluid in the metal are excited when light strikes the surface of the nanostructure. These excitations, known as plasmons, can have evanescent electromagnetic fields that are orders of magnitude larger than the incident electromagnetic field. The largest field enhancements often occur in nanogaps between plasmonically active nanostructures(3,4), but it is extremely challenging to measure the fields in such gaps directly. These enhanced fields have applications in surface-enhanced spectroscopies(5-7), nonlinear optics(1,8-10) and nanophotonics(11-15). Here we show that nonlinear tunnelling conduction between gold electrodes separated by a subnanometre gap leads to optical rectification, producing a d.c. photocurrent when the gap is illuminated. Comparing this photocurrent with low-frequency conduction measurements, we determine the optical frequency voltage across the tunnelling region of the nanogap, and also the enhancement of the electric field in the tunnelling region, as a function of gap size. The measured field enhancements exceed 1,000, consistent with estimates from surface-enhanced Raman measurements(16-18). Our results highlight the need for more realistic theoretical approaches that are able to model the electromagnetic response of metal nanostructures on scales ranging from the free-space wavelength, lambda, down to similar to lambda/1,000, and for experiments with new materials, different wavelengths and different incident polarizations.