One-counter nets (OCN) are Petri nets with exactly one unbounded place. They are equivalent to a subclass of one-counter automata with only a weak test for zero. We show that weak simulation preorder is decidable for OCN and that weak simulation approximants do not converge at level ω, but only at ω 2 . In contrast, other semantic relations like weak bisimulation are undecidable for OCN [17] , and so are weak (and strong) trace inclusion (Sec. 7).