Ionospheric tomography is viewed as an inverse problem, the direct problem of which is linear. The total electron content (TEC) is the integral of the electron density along rays from a satellite to a set of receivers on the ground. If the continuous ionosphere is replaced by a discrete set of finite-sized cells, in each of which the density is constant, then the integrals become simple sums of density times ray lengths in each cell. The equation of the direct problem is then a simple matrix equation. Since the electron density distribution is positive and normalizable, it is isomorphic to a probability distribution for which the maximum entropy method (MaxEnt) yields a unique solution which uses all available information and is maximally noncommittal with respect to all unavailable information. The ionospheric problem is distinct from other forms of tomography in that the rays from the satellite to the receivers are all nearly vertical; horizontal rays which would carry information on the average vertical profile are not available. Without such information the problem cannot be solved; but the nearly vertical rays do carry some such information, although very weakly. The present method is able to extract the required vertical profile for three different test cases, as follows. An arbitrary vertical profile is assumed; for convenience a linear combination of orthonormalized simple Chapman profiles is used; the coefficients of these functions are found using a nonlinear optimization procedure in conjunction with repeated MaxEnt calculations. These test results are unique: There is enough information in the simulated TECs to determine a unique reconstruction in all three test cases. No artifacts appear in the reconstructions, and the resolution is excellent.