We report on observations of very high energy gamma rays from the shell-type supernova remnant (SNR) Cassiopeia A with the Very Energetic Radiation Imaging Telescope Array System stereoscopic array of four imaging atmospheric Cherenkov telescopes in Arizona. The total exposure time for these observations is 22 hr, accumulated between September and November of 2007. The gamma-ray source associated with the SNR Cassiopeia A was detected above 200 GeV with a statistical significance of 8.3 sigma. The estimated integral flux for this gamma-ray source is about 3% of the Crab-Nebula flux. The photon spectrum is compatible with a power law dN/dE proportional to E-Gamma with an index Gamma = 2.61 +/- 0.24(stat) +/- 0.2(sys). The data are consistent with a point-like source. We provide a detailed description of the analysis results and discuss physical mechanisms that may be responsible for the observed gamma-ray emission.