  The $(4,0)$ theory in six dimensions is an exotic theory of supergravity that
has been argued to emerge as the strong coupling limit of theories having $N=8$
supergravity as their low energy effective theory in five spacetime dimensions.
It has maximal supersymmetry and is superconformal. Very little is known about
this intriguing theory. While the spectrum of fields occurring in its
description has been given and their equations of motion in the absence of
interactions have been written down, no action principle has been formulated,
even in the free case. We extend here previous analyses by writing explicitly
the action of the free $(4,0)$ theory from which the equations of motion
derive. The variables of the variational principle are prepotentials adapted to
the self-duality properties of the fields. The "exotic gravitini", described by
chiral fermionic two-forms, are given special attention. The supersymmetry
transformations are written down and the invariance of the action is explicitly
proven. Even though the action is not manifestly covariant, the symmetry
transformations are shown to close according to the $(4,0)$-extended Poincar\'e
supersymmetry algebra. We also discuss exotic supergravity models with fewer
supersymmetries. Remarks on dimensional reduction close the paper.
