The thermodynamics and structural behavior of a new type of transient network are reported. The network is obtained by adding in a solution of entangled surfactant wormlike micelles a telechelic triblock copolymer whose hydrophobic ends anchor into the micelles and whose hydrophilic tails are swollen in the aqueous solvent and reversibly link the entangled cylindrical micelles. For comparison, we have also studied the same surfactant system decorated with an amphiphilic diblock copolymer which corresponds exactly to a triblock telechelic copolymer cut into two identical diblock copolymers. We find that the addition of telechelic polymers induces an effective attraction of entropic origin between the surfactant that may result, in equilibrium, in the coexistence of a dilute phase and a connected network, as predicted by theory. Small-angle neutron scattering experiments show first that the locally cylindrical structure of the micelles is maintained upon copolymer addition over a wide range of copolymer-to-surfactant molar ratio. On the other hand, the addition of telechelic polymer correlates with (i) the emergence of a broad peak in the structure factor at a finite scattering vector, q, due to the short-range steric repulsion induced by the polymer between the cylinders, and (ii) the large rise of the scattered intensity at low q's (for small enough surfactant concentration), which indicates an increase of the osmotic compressibility of the solution due to the effective attractive interaction between cylinders provoked by the telechelic polymers linking them.