Until now the methods of integration for periodic and open Toda lattice were absolutely unrelated to each other. The solution of the periodic Toda lattice is expressed in terms of theta functions of the spectral curve, associated with the auxiliary spectral problem for an infinite periodic Jacobi matrix. For the open Toda the solution of equations of motion was obtained by Moser with the help of the inverse spectral problem for a finite Jacoby matrix which was solved by Stieltjes more than a hundred years ago using continuous fractions. We introduce a spectral curve for the open Toda lattice and demonstrate that algebro--geometrical approach based on the concept of Baker--Akhiezer function can be used for solution of the inverse spectral problem for a finite Jacoby matrix which is different from the classical Stieltjes' solution. Recent interest in this problem is due to connections of the Toda lattice with Seiberg-Witten theory of supersymmetric SU(N) gauge theory. It turns out that open Toda lattice describes weak-coupling limit of this theory.