  Scintillating CaWO$_4$ single crystals are a promising multi-element target
for rare-event searches and are currently used in the direct Dark Matter
experiment CRESST (Cryogenic Rare Event Search with Superconducting
Thermometers). The relative light output of different particle interactions in
CaWO$_4$ is quantified by Quenching Factors (QFs). These are essential for an
active background discrimination and the identification of a possible signal
induced by weakly interacting massive particles (WIMPs). We present the first
precise measurements of the QFs of O, Ca and W at mK temperatures by
irradiating a cryogenic detector with a fast neutron beam. A clear energy
dependence of the QF of O and, less pronounced, of Ca was observed for the
first time. Furthermore, in CRESST neutron-calibration data a variation of the
QFs among different CaWO$_4$ single crystals was found. For typical CRESST
detectors the QFs in the region-of-interest (10-40$\,$keV) are
$QF_O^{ROI}=(11.2{\pm}0.5)\,$%, $QF_{Ca}^{ROI}=(5.94{\pm}0.49)\,$% and
$QF_W^{ROI}=(1.72{\pm}0.21)\,$%. The latest CRESST data (run32) is reanalyzed
using these fundamentally new results on light quenching in CaWO$_4$ having
moderate influence on the WIMP analysis. Their relevance for future CRESST runs
and for the clarification of previously published results of direct Dark Matter
experiments is emphasized.
