This paper is concerned with the reconstruction of an unknown impedance p(x) in the Sturm-Liouville problem with Dirichlet boundary conditions, when only a finite number of eigenvalues are known. The problem is transformed into a system of nonlinear equations. A solution of this system is enclosed in an interval vector by an interval Newton's method. From the interval vector, an interval function [p](x) is constructed that encloses an impedance p(x) corresponding to the prescribed eigenvalues. To make this numerical existence proof rigorous, all discretization and roundoff errors have to be taken into account in the computation.