An ionospheric tomography system consists of a satellite and several ground stations located in a line under the path of the satellite. The data collected at the ground stations are the integrals of electron density along many paths between the ground stations and the satellite. From this data an image of electron density in the plane defined by the satellite orbit and the ground stations can be reconstructed using tomographic techniques. However, the data obtained from an ionospheric tomography system is not complete, so a priori information must be used in the reconstruction algorithm in order to obtain a useful solution. The orthogonal decomposition algorithm (ODA) provides a way of incorporating a priori information into the reconstruction. However, due to problems with numerical conditioning, ODA by itself does not yield a solution. This paper presents a new algorithm for ionospheric tomography reconstruction called the recursive correction method (RCM). RCM is a fast, stable, iterative algorithm that takes advantage of the special structure of the ionospheric tomography problem. This paper will also present convergence analysis of RCM, comparison between RCM and algebraic reconstruction technique (ART), and an example using simulated data.