The self-assembly of organic molecules on surfaces is a promising approach for the development of nanoelectronic devices(1,2). Although a variety of strategies have been used to establish stable links between molecules(2-11), little is known about the electrical conductance of these links. Extended electronic states, a prerequisite for good conductance, have been observed for molecules adsorbed on metal surfaces(12-16). However, direct conductance measurements through a single layer of molecules are only possible if the molecules are adsorbed on a poorly conducting substrate. Here we use a nanoscale four-point probe(17) to measure the conductivity of a self-assembled layer of cobalt phthalocyanine on a silver-terminated silicon surface as a function of thickness. For low thicknesses, the cobalt phthalocyanine molecules lie flat on the substrate, and their main effect is to reduce the conductivity of the substrate. At higher thicknesses, the cobalt phthalocyanine molecules stand up to form stacks and begin to conduct. These results connect the electronic structure and orientation of molecular monolayer and few-layer systems to their transport properties, and should aid in the rational design of future devices.