  We present a new algorithm to compute the integral closure of a reduced
Noetherian ring in its total ring of fractions. A modification, applicable in
positive characteristic, where actually all computations are over the original
ring, is also described. The new algorithm of this paper has been implemented
in Singular, for localizations of affine rings with respect to arbitrary
monomial orderings. Benchmark tests show that it is in general much faster than
any other implementation of normalization algorithms known to us.
