  We discuss the classical and quantum properties of non-local modified
Gauss-Bonnet gravity in de Sitter space, using its equivalent representation
via string-inspired local scalar-Gauss-Bonnet gravity with a scalar potential.
A classical, multiply de Sitter universe solution is found where one of the de
Sitter phases corresponds to the primordial inflationary epoch, while the other
de Sitter space solution--the one with the smallest Hubble rate--describes the
late-time acceleration of our universe. A Chameleon scenario for the theory
under investigation is developed, and it is successfully used to show that the
theory complies with gravitational tests. An explicit expression for the
one-loop effective action for this non-local modified Gauss-Bonnet gravity in
the de Sitter space is obtained. It is argued that this effective action might
be an important step towards the solution of the cosmological constant problem.
