Maximum likelihood (ML) decoding of linear block codes can be considered as an integer linear programming (ILP). Since it is an NP-hard problem in general, there are many researches about the algorithms to approximately solve the problem. One of the most popular algorithms is linear programming (LP) decoding proposed by Feldman et al. LP decoding is based on the LP relaxation, which is a method to approximately solve the ILP corresponding to the ML decoding problem. Advanced algorithms for solving ILP (approximately or exactly) include cutting-plane method and branch-and-bound method. As applications of these methods, adaptive LP decoding and branch-and-bound decoding have been proposed by Taghavi et al. and Yang et al., respectively. Another method for solving ILP is the branch-and-cut method, which is a hybrid of cutting-plane and branch-and-bound methods. The branch-and-cut method is widely used to solve ILP, however, it is unobvious that the method works well for the ML decoding problem. In this paper, we show that the branch-and-cut method is certainly effective for the ML decoding problem. Furthermore the branch-and-cut method consists of some technical components and the performance of the algorithm depends on the selection of these components. It is important to consider how to select the technical components in the branch-and-cut method. We see the differences caused by the selection of those technical components and consider which scheme is most effective for the ML decoding problem through numerical simulations.