  This paper classifies toric Fano 3-folds with singular locus { 1/k(1,1,1) }
for any positive integer k, building on the work of Batyrev and
Watanabe-Watanabe. This is achieved by completing an equivalent problem in the
language of Fano polytopes. Furthermore we identify birational relationships
between entries of the classification. For a fixed value k>4, there are exactly
two such Fano 3-folds linked by a blow-up in a torus-invariant line.
