We propose a dedicated mission in low Earth orbit to test the gravitational interaction and place improved limits on the atmospheric density at the altitudes of interest. A very simple spacecraft would be placed in a dawn-dusk eccentric orbit between 500 and 1200 km. It should be tracked with high precision, and an on-board accelerometer would measure the non-gravitational accelerations acting on it. The precise orbit determination is expected to provide estimates of general relativistic predictions and of geophysical parameters, while the accelerometer data are fundamental in constraining the atmospheric density. Along with the scientific objectives, a basic mission configuration is described and discussed, highlighting the main issues to be explored. © 2014 IEEE.