This is the second paper in a series describing a numerical implementation of the conformal Einstein equation. This paper deals with the technical details of the numerical code used to perform numerical time evolutions from a `minimal' set of data. We outline the numerical construction of a complete set of data for our equations from a minimal set of data. The second- and the fourth-order discretizations, which are used for the construction of the complete data set and for the numerical integration of the time evolution equations, are described and their efficiencies are compared. By using the fourth-order scheme we reduce our computer resource requirements - with respect to memory as well as computation time - by at least two orders of magnitude as compared to the second-order scheme.