For a hypergraph , define the minimum positive -degree to be the largest integer such that every -set which is contained in at least one edge of is contained in at least edges. For and , we prove that for -vertex -intersecting -graphs with , the unique hypergraph with the maximum number of edges is the hypergraph consisting of every edge which intersects a set of size in at least vertices provided is sufficiently large. This generalizes work of Balogh, Lemons, and Palmer who proved this for , as well as the Erdős-Ko-Rado theorem when .