  We argue that the Lagrangian for gravity should remain bounded at large
curvature, and interpolate between the weak-field tested Einstein-Hilbert
Lagrangian L_EH = R /16 pi G and a pure cosmological constant for large R with
the curvature-saturated ansatz L_cs=L_EH/ \sqrt{1+l^4 R^2}, where l is a length
parameter expected to be a few orders of magnitude above the Planck length. The
curvature-dependent effective gravitational constant defined by dL/dR = 1/16 pi
G_eff is G_eff = G \sqrt{1+l^4 R^2}^{3}, and tends to infinity for large $R$,
in contrast to most other approaches where G_eff-> 0. The theory possesses
neither ghosts nor tachyons. In a curvature-saturated cosmology, the
coordinates with ds^2 = a^2 [da^2/B(a) - dx^2 - dy^2- dz^2] are most convenient
since the curvature scalar becomes a linear function of $B(a)$. Solutions with
a big-bang singularity have a much milder behavior of the curvature than in
Einstein's theory. In synchronized time, the metric is given by ds^2 = dt^2 -
t^{6/5(dx^2 + dy^2+ dz^2). On the technical side we show that two different
conformal transformations make L_cs asymptotically equivalent to the
Gurovich-ansatz L= | R |^{4/3} on the one hand, and to Einstein's theory with a
minimally coupled scalar field with self-interaction on the other.
