  Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{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
Erd\H os-Ko-Rado theorem which corresponds to the case $k=1$. Our proof is
based on the delta-system method.
