A formalism capable of handling the first step of Parisi's replica symmetry breaking (RSB) in finite-connectivity models is introduced. The emerging order parameter is claimed to be a probability distribution over the space of field distributions (or, equivalently magnetization distributions) inside the cluster of states. As an application to the study of optimisation problems, the ground-state properties of the random 3-Satisfiability problem are investigated and we present a first and exact RSB solution improving replica symmetric results.