We introduce an extension of the finite Toda lattice, depending on a choice of time independent parameters alpha(0),alpha(1),...,alpha(N-1) is an element of(-infinity, 0), which also includes the finite relativistic Toda lattice as a limit case and comes from the theory of orthogonal rational functions. Generalizing the method Moser used in the case of the finite Toda lattice, we solve this system of nonlinear differential equations for some initial data with the aid of a spectral transform. In particular we study a generalized eigenvalue problem of a pair of matrices (J, I + JD) where J is a symmetric Jacobi matrix and D = diag(alpha(0)(-1),alpha(1)(-1),...,alpha(N-1)(-1).) The inverse spectral transform is described in terms of terminating continued fractions. Finally we compute the time evolution of the spectral data.