In this article, we introduce the concept of the connected edge fixing edge-to-edge geodetic number gcefee(G)for an edge eof a graph G. The connected edge fixing edge-to-edge geodetic number of certain classes of graphs including path, cycles, trees, complete graphs are studied. Connected graphs of size q with gcefee(G) = q – 1 are characterized. It is shown that for a positive integers r, dand ℓwith r < ???? < 2????, there exists a connected graph G with rad(G) = r, diam(G) = d and gcefee(G) = ℓ or ℓ − 1 for some e  E(G).