A graph is said to be edge-superconnected if each minimum edge-cut consists of all the edges incident with some vertex of minimum degree. A graph G is said to be a {d, d + 1}- semiregular graph if all its vertices have degree either d or d+1. A smallest {d,d+1}-semiregular graph G with girth g is said to be a ({d, d+1}; g)-cage.We show that every ({d, d+1}; g)-cage with odd girth g is edge-superconnected.