  The CRESST experiment is a direct dark matter search which aims to measure
interactions of potential dark matter particles in an earth-bound detector.
With the current stage, CRESST-III, we focus on a low energy threshold for
increased sensitivity towards light dark matter particles. In this manuscript
we describe the analysis of one detector operated in the first run of
CRESST-III (05/2016-02/2018) achieving a nuclear recoil threshold of 30.1eV.
This result was obtained with a 23.6g CaWO$_4$ crystal operated as a cryogenic
scintillating calorimeter in the CRESST setup at the Laboratori Nazionali del
Gran Sasso (LNGS). Both the primary phonon/heat signal and the simultaneously
emitted scintillation light, which is absorbed in a separate
silicon-on-sapphire light absorber, are measured with highly sensitive
transition edge sensors operated at ~15mK. The unique combination of these
sensors with the light element oxygen present in our target yields sensitivity
to dark matter particle masses as low as 160MeV/c$^2$.
