Abstract : Understanding the nature of non-baryonic dark matter is one of the most fascinating and challenging goals of modern physics. WIMP (Weak Interactive Massive Particle) direct detection experiments such as EDELWEISS test the hypothesis that dark matter is made up of particles (neutralino) predicted by the supersymmetry and having an interaction cross-section with the nucleon of about 10-8 picobarn. For that purpose, EDELWEISS uses ultrapure cryogenic germanium detectors combining ionisation and heat measurements. This allows the discrimination between nuclear recoils as expected for WIMPs and the electronic recoils induced by the particles of the radioactive background. The main limitation of this technique comes from events close to the detector surface that mimic nuclear recoils. The solution presented here is based on a coplanar grid technique for surface event identification with ionization signals. We present the principle of this type of detector, the fabrication of a prototype and its study: tests with different radioactive sources in a surface laboratory and in the site of the experiment in the Modane underground laboratory (LSM). The experimental results are analysed in details and compared to a modelization of the signals expected for the different event populations. The obtained gamma and surface event (beta) rejection demonstrates the possibility to reach at least 10-8 picobarn in the frame of the EDELWEISS II experiment which chose to use this type of detector for the future.