  A singularity is said to be weakly-exceptional if it has a unique purely log
terminal blow up. This is a natural generalization of the surface singularities
of types $D_{n}$, $E_{6}$, $E_{7}$ and $E_{8}$. Since this idea was introduced,
quotient singularities of this type have been classified in dimensions up to
most $4$. This note extends that classification to dimension $5$.
