CERN-LEP. Measurement of the cross sections for exclusive RHO+ RHO- production in two photon collisions resulting from E+ E- interactions at c.m. energies around the Z-pole (89-93 GeV) and at higher energies (161 to 209 GeV). The data come from a sample with luminosity 148.7 pb-1 around the Z-pole and 706.0 pb-1 at the higher energies. The production cross section is determined as a function of Q**2 and W.