  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.
