This paper covers some key aspects of the implementation and testing of the ACRITH (``High-Accuracy Arithmetic'') facility in the IBM 4361. It also includes a brief description of its definition in the IBM System/370 Architecture RPQ (IBM Publication SA22-7093). The ACRITH facility consists of five new floating-point operations with controlled rounding. Besides the four basic operations (+, -, *, /) the scalar product of two vectors has been architectured. The theory and many details of this paper are neither bound to IBM System/370 architecture nor to hexadecimal number systems. They apply also to other architectures, for example binary or decimal.