  The Bogoliubov-Fr\"ohlich Hamiltonian models the interaction of an impurity
with the excitations of a Bose-Einstein condensate. It has been observed that
the dependence of the ground state energy on the ultraviolet cutoff differs
significantly from what would be expected from similar well-known models. We
give a detailed explanation of this UV behaviour, and provide an explicit
representation of the renormalised Hamiltonian.
