The electrical conduction mechanism in the film plane of ultrathin, evaporated metal films was investigated. These films consist of a planar array of many small discrete islands. The conduction process consists of, first, charge carrier creation which is thermally activated and involves charge transfer between initially neutral particles, and, second, the drift velocity of these charges in an applied field. Charge transfer between particles occurs by tunneling. The following features were predicted and can be verified experimentally: the conductivity depends exponentially on reciprocal temperature, and it should be independent of field at low fields. Deviations from the exponential temperature dependence can be understood in terms of a spectrum of activation energies, while deviations from Ohm's law at high fields can be explained readily in terms of a field dependent activation energy.