The low-viscosity layer in the upper mantle, the asthenosphere, is a requirement for plate tectonics. The seismic low velocities and the high electrical conductivities of the asthenosphere are attributed either to subsolidus, water-related defects in olivine minerals or to a few volume per cent of partial melt, but these two interpretations have two shortcomings. First, the amount of water stored in olivine is not expected to be higher than 50 parts per million owing to partitioning with other mantle phases(including pargasite amphibole at moderate temperatures) and partial melting at high temperatures9. Second, elevated melt volume fractions are impeded by the temperatures prevailing in the asthenosphere, which are too low, and by the melt mobility, which is high and can lead to gravitational segregation. Here we determine the electrical conductivity of carbon-dioxide-rich and water-rich melts, typically produced at the onset of mantle melting. Electrical conductivity increases modestly with moderate amounts of water and carbon dioxide, but it increases drastically once the carbon dioxide content exceeds six weight per cent in the melt. Incipient melts, long-expected to prevail in the asthenosphere, can therefore produce high electrical conductivities there. Taking into account variable degrees of depletion of the mantle in water and carbon dioxide, and their effect on the petrology of incipient melting, we calculated conductivity profiles across the asthenosphere for various tectonic plate ages. Several electrical discontinuities are predicted and match geophysical observations in a consistent petrological and geochemical framework. In moderately aged plates (more than five million years old), incipient melts probably trigger both the seismic low velocities and the high electrical conductivities in the upper part of the asthenosphere, whereas in young plates, where seamount volcanism occurs, a higher degree of melting is expected.