In spite of the successes of observational astro- and particle physics and cosmology very much of the universe remains unknown. The Standard Model of particle physics is a theory describing the electromagnetic, weak, and strong nuclear interactions, as well as classifying all the subatomic particles known. But there is overwhelming evidence, that all the known particles, the ordinary (baryonic) matter, the building blocks of planets, stars and ourselves, only make up about 4.9% of the energy content of the universe. The standard model of cosmology (CDM) indicates that the total mass-energy of the universe contains beside the 4.9% ordinary matter two other components: 26.8% dark matter and 68.3% dark energy. The accelerating expansion of the Universe is the result of the effect of the dark energy with its most simple form given by a cosmological constant in Einstein's Equation. Dark matter is an unidentified type of matter that is not accounted for by dark energy and neutrinos and is generally believed to be a non-relativistic, charge neutral and non-baryonic new form of matter. Although dark matter has not been directly observed yet, its existence and properties are inferred from its gravitational effects such as the motions of visible matter, gravitational lensing, its influence on the universe's large-scale structure, and its effects in the cosmic microwave background. Thus the search for Dark Matter is the search for physics beyond the standard model. Although the nature of dark matter is yet unknown, its presence is crucial to understanding the future of the universe. The CRESST experiment is searching for direct evidence in the form of a nuclear recoil induced on a scintillating CaWO4 crystal by a dark matter particle, and is installed and taking data underground at Laboratory Nazionali del Gran Sasso (LNGS) in Italy. While both, dark energy and dark matter, have not been detected directly, a class of dark matter particles that interact only via gravity and the weak force, referred to asWeakly Interacting Massive Particles (WIMPs), has been established as the leading candidate among the dark matter community. For this thesis a special model of dark matter was studied, namely the dark photon. This thesis provides a detailed description of the calculation of the 90% upper limit on the dark photon kinetic mixing based on data from the second phase of the CRESST experiment. The analysis was carried out in a frequentist approach based on the (unbinned) maximum-likelihood method and likelihood ratios. To make a statement about the calculated result and its quality, the used algorithm had to be tested, what was done with Monte Carlo simulations (pseudo data).