  The direct dark matter search experiment CRESST uses scintillating CaWO$_4$
single crystals as targets for possible WIMP scatterings. An intrinsic
radioactive contamination of the crystals as low as possible is crucial for the
sensitivity of the detectors. In the past CaWO$_4$ crystals operated in CRESST
were produced by institutes in Russia and the Ukraine. Since 2011 CaWO$_4$
crystals have also been grown at the crystal laboratory of the Technische
Universit\"at M\"unchen (TUM) to better meet the requirements of CRESST and of
the future tonne-scale multi-material experiment EURECA. The radiopurity of the
raw materials and of first TUM-grown crystals was measured by ultra-low
background $\gamma$-spectrometry. Two TUM-grown crystals were also operated as
low-temperature detectors at a test setup in the Gran Sasso underground
laboratory. These measurements were used to determine the crystals' intrinsic
$\alpha$-activities which were compared to those of crystals produced at other
institutes. The total $\alpha$-activities of TUM-grown crystals as low as 1.23
$\pm$ 0.06 mBq/kg were found to be significantly smaller than the activities of
crystals grown at other institutes typically ranging between ~15 mBq/kg and ~35
mBq/kg.
