A large amount (about three solar masses) of cold (18 K) dust in the prototypical type II supernova remnant Cassiopeia A was recently reported. It was concluded that dust production in type II supernovae can explain how the large quantities (approximately 10(8) solar masses) of dust observed in the most distant quasars could have been produced within only 700 million years after the Big Bang. Foreground clouds of interstellar material, however, complicate the interpretation of the earlier submillimetre observations of Cas A. Here we report far-infrared and molecular line observations that demonstrate that most of the detected submillimetre emission originates from interstellar dust in a molecular cloud complex located in the line of sight between the Earth and Cas A, and is therefore not associated with the remnant. The argument that type II supernovae produce copious amounts of dust is not supported by the case of Cas A, which previously appeared to provide the best evidence for this possibility.