In recent years, there has been considerable interest in the transformative potential of additive manufacturing (AM) since it allows for producing highly customizable and complex components while reducing lead times and costs. The rise of AM for traditional and new business models enforces the need for efficient planning procedures for AM facilities. In this area, the assignment and sequencing of components to be built by an AM machine, also called a 3D printer, is a complex challenge combining two combinatorial problems: The first decision involves the grouping of parts into production batches, akin to the well-known bin packing problem. Subsequently, the second problem pertains to the scheduling of these batches onto the available machines, which corresponds to a parallel machine scheduling problem. For minimizing makespan, this paper proposes a new branch-and-cut algorithm for integrated planning for unrelated parallel machines. The algorithm is based on combinatorial Benders decomposition: The scheduling problem is considered in the master problem, while the feasibility of an obtained solution with respect to the packing problem is checked in the sub-problem. Current state-of-the-art techniques are extended to solve the orthogonal packing with rotation and used to speed up the solution of the sub-problem. Extensive computational tests on existing and new benchmark instances show the algorithm’s superior performance, improving the makespan by 18.7% on average, with improvements reaching up to 97.6% for large problems compared to an existing integrated mixed-integer programming model.