Abstract: We study a method for approximating a potential $q(x)$ in $$-y''+q(x)y=\lambda y, y(0)=y(\pi)=0$$ from finite spectral data. When the potential is symmetric, the data are the first M Dirichlet eigenvalues. In the general case, the first M terminal velocities are also specified. A centred finite-difference scheme reduces the inverse Sturm-Liouville problem to a matrix inverse eigenvalue problem. Our approach is motivated by the work of Paine, de Hoog and Anderssen, who investigated the discrepancy between continuous and matrix eigenvalues under finite differences. Our modified Newton scheme is based on choosing the number of interior mesh points in the discretization to be 2M. The modified Newton scheme is shown to be convergent for both the case of a symmetric and a general potential. Some numerical experiments are given.