From 1979 to 2001, temperatures observed globally by the mid-tropospheric channel of the satellite-borne Microwave Sounding Unit (MSU channel 2), as well as the inferred temperatures in the lower troposphere, show only small warming trends of less than 0.1K per decade (refs 1-3). Surface temperatures based on in situ observations however, exhibit a larger warming of ~0.17K per decade (refs 4, 5), and global climate models forced by combined anthropogenic and natural factors project an increase in tropospheric temperatures that is somewhat larger than the surface temperature increase. Here we show that trends in MSU channel 2 temperatures are weak because the instrument partly records stratospheric temperatures whose large cooling trend offsets the contributions of tropospheric warming. We quantify the stratospheric contribution to MSU channel 2 temperatures using MSU channel 4, which records only stratospheric temperatures. The resulting trend of reconstructed tropospheric temperatures from satellite data is physically consistent with the observed surface temperature trend. For the tropics, the tropospheric warming is ~1.6 times the surface warming, as expected for a moist adiabatic lapse rate.