For /sup 6/Li beam energies of 2 MeV, ..cap alpha..-..cap alpha.. coincidence measurements show a large, narrow peak due to the direct reaction 2/sup 6/Li..-->..3..cap alpha.., where both /sup 6/Li nuclei contribute one deuteron to make the third ..cap alpha.. particle. The large volume of three-body phase space allows the various parts of the reaction process to be separated from each other. Excellent fits to the data were obtained using a three-body, extended-range, distorted-wave Born-approximation model. Elastic scattering potentials for the calculation were generated by the double-folding technique in combination with /sup 6/Li+/sup 6/Li elastic scattering measurements for /sup 6/Li beam energies from 2 to 5.5 MeV.