The hypothesis that the M discontinuity is a phase change corresponding to the transition between basalt and eclogite offers a promising explanation for the occurrence of great thicknesses of shallow-water sedimentary deposits. Although the hypothesis encounters serious difficulties if it is applied on a worldwide basis to continents and oceans alike, these difficulties are much less severe if it is applied only to parts of the continents. In any case, the hypothesis deserves consideration because of its important consequences. A numerical method is presented in this report for determining the history of a sedimentary basin, based on the assumption that such a phase change is present. This method, like the one developed by van de Lindt, takes into account isostasy, which was neglected in earlier work on the same problem by MacDonald and Ness. For models in which clastic deposition was assumed and the initial basin depth was 1.5 km, 7 to 10 km of sediments were deposited, depending on deposition rates and other parameters. With an initial depth of only 500 meters, 4 km of clastics were deposited or 6 km of limestone. For the models studied, the time spans of deposition were as long as 40 m.y. for clastic sediments and 200 m.y. or longer for limestone or mixed clastics and limestone. One model exhibited a cyclic alternation of periods of deposition and erosion. During the second cycle of deposition, 3 km of mixed clastics and limestone were deposited, altogether the water depth never exceeded 170 meters. In all the models deposition was followed by uplift. Study of the uplift cycle was handicapped by limitations in the computer program, but the results indicate that uplift cycles lasting as long as 200 to 300 m.y. can readily be explained. Attempts to devise a model to duplicate the history of the Appalachian miogeosyncline resulted in a fairly good match for total thickness and time span of deposition, but this required either an initial water depth of 1.5 km, which is clearly too large, or rather high values for the density and thermal conductivity of the sediments. In view of the many simplifications imposed by the numerical method, however, the results are considered favorable to the phase change hypothesis.