High-brightness ultra-cold muon beams have many applications in particle and solid state physics. One way to produce such beam from the secondary surface muon beam is by using a combination of crossed electric and magnetic fields and a gas density gradient. This thesis was concerned with the development of the transversal compression stage for that beam. It was done at ETH Ziirich, in the group for Precision physics at Low Energy. The simulations of the electric fields and muon drift in the cell for transverse compression were done and it was shown that it is possible to compress the muon beam with combination of the vertical temperature gradient and crossed electric and magnetic fields. Furthermore, various experimental tests were done to investigate the mechanical stability of the materials used for the construction of the cell at low temperatures. It was found that Kapton foil, 3D printed plastic and quartz are suitable for this purpose. The permeability of helium through a Kapton foil used to enclose the cell was measured. The obtained value for room temperature is (2.55 ± 0.02) barrer, which is in good agreement with literature. It was demonstrated that the permeability constant obeys Arrhenius' law for temperatures ranging from 155 K to 300 K. Thus, the leak rate of helium will be sufficiently low at low temperatures. F inally, the performance of the scintillators read out with wavelength shifting fibers and Geiger-mode avalanche photodiodes for positron detection was studied. The light yield of such detector setup was found to be too low to clearly distinguish signal from the noise, which means that this still needs to be improved. However, it should be possible to test the feasibility of the transverse muon compression during the beam time in December 2014, since the remaining challenges are only of a technical nature.