Let D be a finite and simple digraph with vertex set V(D).The minimum degree δ of a digraph D is defined as the minimum value of its out-degrees and its in-degrees.If D is a digraph with minimum degree δ and edge-connectivity λ,then λ ≤ δ.A digraph is maximally edge-connected if λ＝δ.A digraph is called super-edge-connected if every minimum edge-cut consists of edges incident to or from a vertex of minimum degree.In this note we show that a digraph is maximally edge-connected or super-edge-connected if the number of arcs is large enough.