In this paper, a level bundle method with non-Euclidean distance is proposed for minimizing constrained convex nonsmooth optimization problems. In the projection subproblem, the Bregman distance is used to replace the classical Euclidean distance, in order that the geometric structure of the feasible set can be taken into account, and therefore the computational efficiency could be improved. Global convergence of the algorithm is proved and the iterative complexity is analyzed.