Electron dynamics at interfaces is a subject of great scientific interest and technological importance. Detailed understanding of such dynamics requires access to the angstrom length scale defining interfaces and the femtosecond time scale characterizing interfacial motion of electrons. In this context, the most precise and general way to remotely measure charge dynamics is through the transient current flow and the associated electromagnetic radiation. Here, we present quantitative measurements of interfacial currents on the subnanometer length and femtosecond time scale by recording the emitted terahertz radiation following ultrafast laser excitation. We apply this method to interlayer charge transfer in heterostructures of two transition metal dichalcogenide monolayers less than 0.7 nm apart. We find that charge relaxation and separation occur in less than 100 fs. This approach allows us to unambiguously determine the direction of current flow, to demonstrate a charge transfer efficiency of order unity, and to characterize saturation effects.