Summary: Let \(\mathcal{H}\) be an \(r\)-uniform hypergraph. The minimum positive co-degree of \(\mathcal{H} \), denoted by \(\delta_{r-1}^+(\mathcal{H})\), is the minimum \(k\) such that if \(S\) is an \((r-1)\)-set contained in a hyperedge of \(\mathcal{H}\), then \(S\) is contained in at least \(k\) hyperedges of \(\mathcal{H}\). For \(r\geq k\) fixed and \(n\) sufficiently large, we determine the maximum possible size of an intersecting \(r\)-uniform \(n\)-vertex hypergraph with minimum positive co-degree \(\delta_{r-1}^+(\mathcal{H})\geq k\) and characterize the unique hypergraph attaining this maximum. This generalizes the Erdős-Ko-Rado theorem which corresponds to the case \(k=1\). Our proof is based on the delta-system method.