Atmospheric density profiles in the stratosphere and mesosphere are determined by means of low cost Global Positioning System (GPS) receivers on in situ rigid falling spheres released from a sounding rocket. Values below an altitude of 80 km are obtained. Aerodynamic drag relates atmospheric densities to other variables such as velocities of spheres, drag coefficients,and reference area.The densities are reconstructed by iterative solution. The calculated density is reasonably accurate, with deviation within 10 % with respect to the European Centre for Medium-range Weather Forecasts ( ECMWF) reference value. The atmospheric temperature and wind profiles are obtained as well, and compared to independent data.