  We resurrect Eddington's proposal for the gravitational action in the
presence of a cosmological constant and extend it to include matter fields. We
show that the Newton-Poisson equation is modified in the presence of sources
and that charged black holes show great similarities with those arising in
Born-Infeld electrodynamics coupled to gravity. When we consider homogeneous
and isotropic space-times we find that there is a minimum length (and maximum
density) at early times, clearly pointing to an alternative theory of the Big
Bang. We thus argue that the modern formulation of Eddington's theory,
Born-Infeld gravity, presents us with a novel, non-singular description of the
Universe.
