Electromagnetic wave propagation in an ionized gas, particularly the ionosphere, has received considerable attention for many years. In the present paper, Riemann invariants for Maxwell’s equations are applied to the inverse problem of determining the electron density of a plasma in a spherical coordinate system. The tangential components of the electromagnetic fields are given on a part of the surface of a sphere (the source, an electric impulse dipole, is located at the center of the sphere), and the reconstruction of the electron density outside the sphere is desired. The structure of the fundamental solution is analyzed. A uniqueness theorem and a stability estimate for the inverse problem are given.