/ AbstractModels 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 $ν$-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$^2$.