  Recently a novel perturbative continuum limit for quantum gravity has been
proposed and demonstrated to work at first order. Every interaction monomial
$\sigma$ is dressed with a coefficient function $f^\sigma_\Lambda(\varphi)$ of
the conformal factor field, $\varphi$. Each coefficient function is
parametrised by an infinite number of underlying couplings, and decays at large
$\varphi$ with a characteristic amplitude suppression scale which can be chosen
to be at a common value, $\Lambda_\text{p}$. Although the theory is
perturbative in couplings it is non-perturbative in $\hbar$. At second order in
perturbation theory, one must sum over all melonic Feynman diagrams to obtain
the particular integral. We show that it leads to a well defined renormalized
trajectory and thus continuum limit, provided it is solved by starting at an
arbitrary cutoff scale $\Lambda=\mu$ which lies in the range
$0<\mu<a\Lambda_\text{p}$ ($a$ some non-universal number). If $\mu$ lies above
this range the resulting coefficient functions become singular, and the flow
ceases to exist, before the physical limit is reached. To this one must add a
well-behaved complementary solution, containing irrelevant couplings determined
uniquely by the first-order interactions, and renormalized relevant couplings.
Even though some irrelevant couplings diverge in the limit
$\Lambda_\text{p}\to\infty$, domains for the underlying relevant couplings can
be chosen such that diffeomorphism invariance will be recovered in this limit,
and where the underlying couplings disappear to be replaced by effective
diffeomorphism invariant couplings.
