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The CRESST-II cryogenic dark matter search aims for the detection of WIMPs via elastic scattering off nuclei in CaWO$_4$ crystals. We present results from a low-threshold analysis of a single upgraded detector module. This module efficiently vetoes low energy backgrounds induced by $\alpha $ -decays on inner surfaces of the detector. With an exposure of 29.35 kg live days collected in 2013 we set a limit on spin-independent WIMP-nucleon scattering which probes a new region of parameter space for WIMP masses below 3 GeV/c$^2$ , previously not covered in direct detection searches. A possible excess over background discussed for the previous CRESST-II phase 1 (from 2009 to 2011) is not confirmed.