  We complete the explicit study of a three-fold divisorial contraction whose
exceptional divisor contracts to a point, by treating the case where the point
downstairs is a singularity of index $n \ge 2$. We prove that if this
singularity is of type c$A/n$ then any such contraction is a suitable weighted
blow-up; and that if otherwise then the discrepancy is $1/n$ with a few
exceptions. Every such exception has an example. Some exceptions allow the
discrepancy to be arbitrarily large, but any contraction in this case is
described as a weighted blow-up of a singularity of type c$D/2$ embedded into a
cyclic quotient of a smooth five-fold. The erratum to the previous paper [14]
is attached.
