  We construct a prequantum 2-Hilbert space for any line bundle gerbe whose
Dixmier-Douady class is torsion. Analogously to usual prequantisation, this
2-Hilbert space has the category of sections of the line bundle gerbe as its
underlying 2-vector space. These sections are obtained as certain morphism
categories in Waldorf's version of the 2-category of line bundle gerbes. We
show that these morphism categories carry a monoidal structure under which they
are semisimple and abelian. We introduce a dual functor on the sections, which
yields a closed structure on the morphisms between bundle gerbes and turns the
category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert
spaces fit various expectations from higher prequantisation. We then extend the
transgression functor to the full 2-category of bundle gerbes and demonstrate
its compatibility with the additional structures introduced. We discuss various
aspects of Kostant-Souriau prequantisation in this setting, including its
dimensional reduction to ordinary prequantisation.
