  A new, Eddington inspired theory of gravity was recently proposed by Banados
and Ferreira. It is equivalent to General Relativity in vacuum, but differs
from it inside matter. This viable, one parameter theory was shown to avoid
cosmological singularities and turns out to lead to many other exciting new
features that we report here. First, for a positive coupling parameter, the
field equations have a dramatic impact on the collapse of dust, and do not lead
to singularities. We further find that the theory supports stable, compact
pressureless stars made of perfect fluid, which provide interesting models of
self-gravitating dark matter. Finally, we show that the mere existence of
relativistic stars imposes a strong, near optimal constraint on the coupling
parameter, which can even be improved by observations of the moment of inertia
of the double pulsar.
