20 pages, 2 tables, 9 figures, 1 appendix.-- PACS nrs.: 25.60.-t; 27.20.+n; 25.70.De; 25.75.Gz.-- Printed version published Sep 5, 2005.Dissociation of He-6 on a lead target has been studied at 240 MeV/u. The four-momenta of the two neutrons in coincidence with alpha-particles have been measured and used to reconstruct the three-body energy and angular correlations in the final state. For the first time, three-body correlations were included in the analysis of the He-6 dissociation data, and compared with calculations assuming a dipole mode for the electromagnetic dissociation. In addition, the experimental data have been analyzed using a series expansion of the final transition amplitude into hyperspherical functions, showing the importance of both nn and alpha n final-state interactions. It is shown that the dissociation process is governed by a transition of one neutron from the p-shell in the 6He ground state to the s-shell in the continuum further indicating that the role of the transition through an intermediate He-5(3/2(-)) state is more important than expected.This work was supported by the BMBF under Contracts 06 DA 115, 06 OF 112 and 06 MZ 174 and by GSI via Hochschulzusammenarbeitsvereinbarungen under Contracts DA RICK, OF ELZ and partly supported by the Polish Committee of Scientific Research under Contract PBZ/PB03/113/09, EC under contract ERBCHGE-CT92-0003, CICYT under contract AEN92-0788-C02-02 (MJGB) and RFBR grant No. 04-02-17251. One of us (B.J.) acknowledges the support through an Alexander von Humbold Research Award.Peer reviewed