Long wavelength time variations of the gravity field detected by means of laser tracking of geodetic satellites provide nowadays the unique opportunity to infer simultaneously the upper and lower mantle viscosity and the mass imbalance between ice accumulation and loss in Antarctica and Greenland. Here we show that mantle viscosity inversion based on stratified viscoelastic Earth models of the Pleistocene deglaciation, on two independent Satellite Laser Ranging (SLR) solutions for the even and odd zonal components of the geopotential and on True Polar Wander (TPW), indicate that present day ice mass loss in Antarctica amounts to about -250 Gt/yr, equivalent to a global sea level rise of 0.7 mm/yr, and that the lower mantle is more viscous than the upper mantle. The odd zonals of the two SLR solutions and TPW indicate two admissible upper mantle viscosities, close to 1020 Pa s and to 1021 Pa s respectively.