We present a proximal bundle method for finding weakly Pareto optimal solutions to constrained nonsmooth programming problems with multiple objectives. The method is a generalization of proximal bundle approach for single objective optimization. The multiple objective functions are treated individually without employing any scalarization. The method is globally convergent and capable of handling several nonconvex locally Lipschitz continuous objective functions subject to nonlinear (possibly nondifferentiable) constraints. Under some generalized convexity assumptions, we prove that the method finds globally weakly Pareto optimal solutions. Concluding, some numerical examples illustrate the properties and applicability of the method.