  Motivated by recent evidence indicating that Quantum Einstein Gravity (QEG)
might be nonperturbatively renormalizable, the exact renormalization group
equation of QEG is evaluated in a truncation of theory space which generalizes
the Einstein-Hilbert truncation by the inclusion of a higher-derivative term
$(R^2)$. The beta-functions describing the renormalization group flow of the
cosmological constant, Newton's constant, and the $R^2$-coupling are computed
explicitly. The fixed point (FP) properties of the 3-dimensional flow are
investigated, and they are confronted with those of the 2-dimensional
Einstein-Hilbert flow. The non-Gaussian FP predicted by the latter is found to
generalize to a FP on the enlarged theory space. In order to test the
reliability of the $R^2$-truncation near this FP we analyze the residual scheme
dependence of various universal quantities; it turns out to be very weak. The
two truncations are compared in detail, and their numerical predictions are
found to agree with a suprisingly high precision. Due to the consistency of the
results it appears increasingly unlikely that the non-Gaussian FP is an
artifact of the truncation. If it is present in the exact theory QEG is
probably nonperturbatively renormalizable and ``asymptotically safe''. We
discuss how the conformal factor problem of Euclidean gravity manifests itself
in the exact renormalization group approach and show that, in the
$R^2$-truncation, the investigation of the FP is not afflicted with this
problem. Also the Gaussian FP of the Einstein-Hilbert truncation is analyzed;
it turns out that it does not generalize to a corresponding FP on the enlarged
theory space.
