We consider a bounded Jacobi operator acting in the space l²(ℕ). We supplement the spectral measure of this operator by a set of finitely many discrete masses (on the real axis outside the convex hull of the support of the operator's spectral measure). In the present paper, we study whether the obtained perturbation of the original operator is compact. For limit-periodic Jacobi operators, we obtain a necessary and sufficient condition on the location of the masses for the perturbation to be compact.