We give a proximal bundle method for minimizing a convex function $f$ over a closed convex set. It only requires evaluating $f$ and its subgradients with an accuracy $\epsilon>0$, which is fixed but possibly unknown. It asymptotically finds points that are $\epsilon$-optimal. When applied to Lagrangian relaxation, it allows for $\epsilon$-accurate solutions of Lagrangian subproblems and finds $\epsilon$-optimal solutions of convex programs.