  A new design of a cryogenic germanium detector for dark matter search is
presented, taking advantage of the coplanar grid technique of event
localisation for improved background discrimination. Experiments performed with
prototype devices in the EDELWEISS II setup at the Modane underground facility
demonstrate the remarkably high efficiency of these devices for the rejection
of low-energy $\beta$, approaching 10$^5$ . This opens the road to investigate
the range beyond 10$^{-8}$ pb in the WIMP-nucleon collision cross-sections, as
proposed in the EURECA project of a one-ton cryogenic detector mass.
