The cryogenic dark matter experiment CRESST-II aims at the direct detection of WIMPs via elastic scattering off nuclei in scintillating CaWO$_4$ crystals. We present a new, highly improved, detector design installed in the current run of CRESST-II Phase 2 with an efficient active rejection of surface-alpha backgrounds. Using CaWO$_4$ sticks instead of metal clamps to hold the target crystal, a detector housing with fully-scintillating inner surface could be realized. The presented detector (TUM40) provides an excellent threshold of ${\sim}\,0.60\,$keV and a resolution of $\sigma\,{\approx}\,0.090$ keV (at 2.60$\,$keV). With significantly reduced background levels, TUM40 sets stringent limits on the spin-independent WIMP-nucleon scattering cross section and probes a new region of parameter space for WIMP masses below 3$\,$GeV/c$^2$. In this paper, we discuss the novel detector design and the surface-alpha event rejection in detail.
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