The Far Infrared and Submillimetre Space Telescope is one of ESA's cornerstone missions in the “horizon 2000” programme. One option to cool the scientific instruments of this satellite is a superfluid helium (He II) cryostat. A trade-off analysis was performed about different sizes and configurations of a cryostat such as to meet the operational lifetime requirement of 2 - 4 years on either a highly elliptical 24 hours orbit or on a far-away from earth orbit. The possible size alternatives for a cryostat using to a maximum extent existing ISO-technology are presented. Upon ESA-decision a final cryostat size for the 24 hours orbit which provides at least 3 years operational lifetime was selected. This cryostat is described and presented.