  Models for light dark matter particles with masses below 1 GeV/c$^2$ are a
natural and well-motivated alternative to so-far unobserved weakly interacting
massive particles. Gram-scale cryogenic calorimeters provide the required
detector performance to detect these particles and extend the direct dark
matter search program of CRESST. A prototype 0.5 g sapphire detector developed
for the $\nu$-cleus experiment has achieved an energy threshold of
$E_{th}=(19.7\pm 0.9)$ eV, which is one order of magnitude lower than previous
results and independent of the type of particle interaction. The result
presented here is obtained in a setup above ground without significant
shielding against ambient and cosmogenic radiation. Although operated in a
high-background environment, the detector probes a new range of light-mass dark
matter particles previously not accessible by direct searches. We report the
first limit on the spin-independent dark matter particle-nucleon cross section
for masses between 140 MeV/c$^2$ and 500 MeV/c$^2$.
