We construct the fermion zero-mode for arbitrary charge one SU(n) calorons
with non-trivial holonomy, both in the finite temperature context (anti-per
iodic boundary conditions in time) and in the Kaluza-Klein compactification
context (periodic boundary conditions in time). The zero-mode is localised
on one of the constituent monopoles and we discuss a relation to the Calli
as index theorem.