  The non-perturbative renormalisation of quantum gravity is investigated
allowing for the metric to be reparameterised along the RG flow, such that only
the essential couplings constants are renormalised. This allows us to identify
a universality class of quantum gravity which is guaranteed to be unitary,
since the physical degrees of freedom are those of general relativity without
matter and with a vanishing cosmological constant. Considering all
diffeomorphism invariant operators with up to four derivatives, only Newton's
constant is essential at the Gaussian infrared fixed point associated to the
linearised Einstein--Hilbert action. The other inessential couplings can then
be fixed to the values they take at the Gaussian fixed point along the RG flow
within this universality class. In the ultraviolet, the corresponding beta
function for Newton's constant vanishes at the interacting Reuter fixed point.
The properties of the Reuter fixed point are stable between the
Einstein--Hilbert approximation and the approximation including all
diffeomorphism invariant four derivative terms in the flow equation. Our
results suggest that Newton's constant is the only relevant essential coupling
at the Reuter fixed point. Therefore, we conjecture that quantum Einstein
gravity, the ultraviolet completion of Einstein's theory of general relativity
in the asymptotic safety scenario, has no free parameters in the absence of
matter and in particular predicts a vanishing cosmological constant.
