The aim of the present work is to understand the origin of the long-standing discrepancy between the EUV/UV-based predictions of the quiet-Sun microwave spectrum and the observed one. We compare accurate measurements of the quiet-Sun microwave brightness temperature (Tb) with theoretical calculations obtained by using the differential emission measure (DEM) of the plasma derived from UV and EUV spectral line intensities observed by the SUMER and CDS instruments on board the Solar and Heliospheric Observatory (SOHO). No agreement can be found between the observed Tb and calculations carried out using the standard DEM curves obtained from the EUV/UV observations. In order to obtain agreement, it is necessary (1) to modify the temperature range in which the DEM is usually defined in order to take into account the presence of an isothermal corona, (2) to separate the contribution of the cell and the network structures in the transition region, and (3) to substitute the EUV/UV-based DEM values at very low temperature (logT<=4.3) with values based on the Vernazza, Avrett, & Loeser model. In the present work we are able to solve a long-standing discrepancy between microwave and EUV/UV results, and we demonstrate the great potential of the simultaneous use of observations in these two spectral ranges.