We present the first complete, exact and efficient C++ implementation of a method for parameterizing the intersection of two implicit quadrics with integer coefficients of arbitrary size. It is based on the near-optimal algorithm recently introduced by Dupont et al.~\cite{dupont03a}. Unlike existing implementations, it correctly identifies and parameterizes all the connected components of the intersection in all the possible cases, returning parameterizations with rational functions whenever such parameterizations exist. In addition, the coefficient field of the parameterizations is either minimal or involves one possibly unneeded square root.