A connected matching in a graph G consists of a set of pairwise disjoint edges whose covered vertices induce a connected subgraph of G. While finding a connected matching of maximum cardinality is a well-solved problem, it is NP-hard to determine an optimal connected matching in an edge-weighted graph, even in the planar bipartite case. We present two mixed integer programming formulations and a sophisticated branch-and-cut scheme to find weighted connected matchings in general graphs. The formulations explore different polyhedra associated to this problem, including strong valid inequalities both from the matching polytope and from the connected subgraph polytope. We conjecture that one attains a tight approximation of the convex hull of connected matchings using our strongest formulation, and report encouraging computational results over DIMACS Implementation Challenge benchmark instances. The source code of the complete implementation is also made available.