  In this paper, we develop a fully discrete entropy preserving
ADER-Discontinuous Galerkin (ADER-DG) method. To obtain this desired result, we
equip the space part of the method with entropy correction terms that balance
the entropy production in space, inspired by the work of Abgrall. Whereas for
the time-discretization we apply the relaxation approach introduced by
Ketcheson that allows to modify the timestep to preserve the entropy to machine
precision. Up to our knowledge, it is the first time that a provable fully
discrete entropy preserving ADER-DG scheme is constructed. We verify our
theoretical results with various numerical simulations.
