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 weighted blowup; and that if otherwise, then $f$ is either a weighted blowup of a singularity of type c $D/2$ embedded into a cyclic quotient of a smooth five-fold, or a contraction with discrepancy $1/n$ , 1, or 2. Every such exceptional case of discrepancy 1 or 2 has an example. The erratum to our previous article [13] appears in the appendix.