We define minimal fusion systems in a way that every non-solvable fusion system has a section which is minimal. Minimal fusion systems can also be seen as analogs of Thompson's N-groups. In this paper, we consider a minimal fusion system F on a finite p-group S that has a unique maximal p-local subsystem containing N_F(S). For an arbitrary prime p, we determine the structure of a certain (explicitly described) p-local subsystem of F. If p=2, this leads to a complete classification of the fusion system F.