  We study the structure of the category of integrable level zero
representations with finite dimensional weight spaces of affine Lie algebras.
We show that this category possesses a weaker version of the finite length
property, namely that an indecomposable object has finitely many simple
constituents which are non-trivial as modules over the corresponding loop
algebra. Moreover, any object in this category is a direct sum of
indecomposables only finitely many of which are non-trivial. We obtain a
parametrization of blocks in this category.
