[en] An apparatus for a test of a basic postulate of the theory of Special Relativity, the isotropy of the speed of light, has been developed. Deviations from the isotropy imply a violation of Lorentz invariance, a symmetry assumed by all established theories of the fundamental forces. Such a signal may provide a glimpse on physics beyond our current theories of the fundamental forces, the General Theory of Relativity and the Standard Modell of particle physics. Since long theoreticians try to unify General Relativity and the Standard Modell within one theory, a grand unified theory (GUT). So far they did not succeed, although promising candidate theories have been developed, e.g. string theories or loop quantum gravity. However, there are hints that Lorentz invariance might not be an exact symmetry of nature, but that deviations are to be expected. This is a strong motivation for tests of Lorentz invariance with increased sensitivity as the one presented within this thesis. We employ, for the first time for a test of the isotropy of the speed of light, monolithic optical resonators fabricated from a glass ceramic with ultra low expansion coefficient (ULE). By means of a monolithic Nd:YAG-laser (λ = 1064 nm) we measure the difference between the resonance frequencies of two orthogonally oriented resonators. The low thermal expansion coefficient reduces the influence of thermal fluctuations on the resonance frequencies, which are a function of the mirror spacing and the speed of light inside the resonators only. The complete optical setup has been put on top of active vibration isolation supports, which strongly damp mechanical vibrations. This improves the short-time stability of the resonators resonance frequencies. This technique is used for the first time in a Speed of Light Isotropy Test (SLIT) experiment. Furthermore, a system for the stabilization of the tilt of the optics breadboard is implemented, based on electromagnetic actuators. This stabilization is necessary, since a tilt of the resonators with respect to the local direction of gravitation leads to a shift of the resonance frequencies. Finally, servo systems have been implemented to stabilize the optical power circulating in the resonators and the temperature of the setup. The complete experiment can be actively rotated by means of an high precision air bearing rotation table, which allows for a considerable increase of the rate of data taking as compared to a stationary system. This allows for a comparatively fast reduction of statistical errors. Another advantage of the active rotation is the relaxation of stability demands for long timescales. A high frequency stability is needed on a timescale of half a rotation, here ∼ 45 s, while for stationary systems it would be 12 hours since one has to rely on the rotation of the earth. With the setup just described we have performed measurements between march 2008 and may 2009 yielding ∼ 135000 rotations distributed over the entire timespan. This data was analyzed according to two different test theories, the Robertson-Mansouri-Sexl theory (RMS) and the Standard Modell extension (SME). Within the RMS theory a single parameter combination (δ - β + 1/2) describes a possible anisotropy. For an isotropic speed of light it equals zero. We determined an upper limit of (δ - β + 1/2) ≤ 8 . 10^-12 corresponding to a relative anisotropy of the speed of light below (1)/(2) vertical stroke Δc(π/2)/c vertical stroke ≤ 6 . 10^-18 (1σ bounds). This value is more than one order of magnitude smaller than the values published so far. Within the framework of the SME we could determine 8 parameters describing a possible violation of the Lorentz invariance by photons. Upper limits for these parameters could be improved considerably compared to the experimental predecessor of the setup and to the values determined by other groups. Parts of this thesis have already been published in Optics Communications: Ch. Eisele, M. Okhapkin, A. Yu. Nevsky and S. Schiller, Optics Communications 281, 1189 (2008). The final results have been published in Physical Review Letters: Ch. Eisele, A. Yu. Nevsky, and S. Schiller, Physical Review Letters 103, 090401 (2009). (orig.)