  Following Davie's example of a Banach space failing the approximation
property [D], we show how to construct a Banach space E which is asymptotically
Hilbertian and fails the approximation property. Moreover, the space E is shown
to be a subspace of a space with an unconditional basis which is ``almost'' a
weak Hilbert space and which can be written as the direct sum of two subspaces
all of whose subspaces have the approximation property.
