By using the recurrence relation between the characteristic polynomials of the lead-ing principal submatrices of a Jacobi matrix, we resolve the Jacobi matrix comple-tion problem with prescribed minimal and maximal eigenvalues. The necessary and su?cient condition for that the problem exists the unique solution is proved. More-over, the recurrence relations of the inserted elements are obtained. Furthermore, based on these results, the inverse eigenvalue problem of the Jacobi matrix with 2n?1 extreme eigenvalues is solved. Finally, two algorithms for the problem are proposed. Numerical examples verify the effectiveness of the proposed algorithms.