The EDELWEISS Dark Matter Search uses low-temperature Ge detectors with heat and ionisation read-out to identify nuclear recoils induced by elastic collisions with WIMPs from the galactic halo. Preliminary results obtained with 320g bolometers are described. After a few weeks of data taking, data accumulated with one of these detectors already allow to reach the upper part of the DAMA region. Prospects for the present run and the second stage of the experiment, EDELWEISS-II, using an innovative reversed cryostat allowing data taking with 100 detectors, are briefly described.