  We study a non-linear modification to General Relativity in which the
standard Einstein-Hilbert action is replaced by a Born-Infeld type action. Also
study us stability issues to judge about viability of this modification. We
establish the conditions that this modification must satisfy for to avoid the
problems associated with the Dolgov-Kawasaki instability, tachyon instability
and negative effective gravitational coupling. The particle content of
gravitational spectrum of the linearized Born-Infeld theory shows the existence
of massless gravitons plus new degree of freedom of 0-spin associated with
$R^2$ term. For a toy model we proved that for appropriate values of the
background curvature, this theory is free of ghost, Dolgov-Kawasaki instability
and tachyon instability also it has a positive effective gravitational
coupling. We find the scalar-tensor theory equivalent at this $f(R)$-theory in
the Einstein frame, and we study some properties of the scalar potential.
