Angular distributions for 1n and 2n transfer are reported for the 6He+65Cu system at E_{lab}=22.6 MeV. For the first time, triple coincidences between alpha particles, neutrons, and characteristic gamma rays from the targetlike residues were used to separate the contributions arising from 1n and 2n transfer. The differential cross sections for these channels, elastic scattering, and fusion were analyzed using a coupled reaction channels approach. The large measured ratio of the 2n-to-1n cross section and the strong influence of 2n transfer on other channels indicate that the dineutron configuration of 6He plays a dominant role in the reaction mechanism.