Given an undirected node-weighted graph, the Maximum-Weight Connected Subgraph problem (MWCS) is to identify a subset of nodes of maximalsum of weights that induce a connected subgraph. MWCS is closely related to the well-studied Prize Collecting Steiner Tree problem and has many applications in different areas, including computational biology, network design and computer vision. The problem is NP-hard and even hard to approximate within a constant factor. In this work we describe an algorithmic scheme for solving MWCS to provable optimality, which is based on preprocessing rules, new results on decomposing an instance into its biconnected and triconnected components and a branch-and-cut approach combined with a primal heuristic. We demonstrate the performance of our method on the benchmark instances of the 11th DIMACS implementation challenge consisting of MWCS as well as transformed PCST instances.