232 hours of data were accumulated from 1997 to 1999, using the HEGRA Stereoscopic Cherenkov Telescope System to observe the supernova remnant Cassiopeia A. TeV γ-ray emission was detected at the 5 σ level, and a flux of (5.8 ± 1.2stat ± 1.2syst) 10-9 ph m-2 s-1 above 1 TeV was derived. The spectral distribution is consistent with a power law with a differential spectral index of -2.5 ± 0.4stat ± 0.1syst between 1 and 10TeV. As this is the first report of the detection of a TeV γ-ray source on the "centi-Crab" scale, we present the analysis in some detail. Implications for the acceleration of cosmic rays depend on the details of the source modeling. We discuss some important aspects in this paper.