Electromagnetic data images mantle regions more conductive than that of dry olivine. There is no doubt that melt is thermodynamically stable and present in the LAB, but how they can impact on mantle electrical conductivity remains debated. In addition, gravitational segregation and fast melt upwelling, being expected if melt fraction exceeds 2 vol. %, is thought to seriously restrict the role of partial melting at the level of the LAB. Petrological studies realized some 30 years ago have shown that peridotites exposed at the P-T-fO2 conditions of the LAB produced H2O and CO2 rich-melts. The segregation of such melts is not expected since they constitute only about 0.5 vol. % of the peridotite, but electrical conductivities of these melts are poorly known. Therefore, electrical conductivity experiments have been performed in piston cylinder on H2O-CO2 rich melts. Different melt compositions have been explored, from carbonated melts to basalts. The effects of chemical compositions and volatiles on these melts have been determined. The electrical conductivity measurements have shown that hydrous carbonated melts are very conductive, and the incorporation of basalt decreases the conductivity. With these new data, a semi-empirical law predicting the conductivity as a function of H2O and CO2 contents has been produced. Based on this law and the electrical conductivity of olivine, 1D conductivity profiles were constructed. With these profiles, the effect of volatiles content (partitioned between the melt and in the solids), melt fractions (mixing law and interconnection of the melt) and different temperature regimes on conductivity are discussed. These calculations are conducted on oceanic and continental settings with different ages. The electrical conductivities of the mantle is thus a powerful tool to track the fundamental process of mantle incipient melting, which is in turn narrowly associated to the cycling of H2O and CO2 in the upper mantle.