  BRST-invariant action of general relativity in the unimodular gauge proposed
by Baulieu is studied without using perturbative expansions. The expression for
the path integral in the unimodular gauge is reduced to a form in which a
functional measure is defined by a norm invariant under Transverse
Diffeomorphism. It is shown that general relativity in the unimodular gauge
with this action and the quantum unimodular gravity are equivalent. It is also
shown that Vacuum Expectation Values (VEVs) of Diff invariant operators in the
unimodular gauge and other gauge such as the harmonic gauge take distinct
values. A path integral for harmonic gauge is found to be gauge equivalent to a
superposition of that for unimodular gauge obtained by performing constant Weyl
transformation of the metric, after a non-dynamical cosmological term is
introduced into the action of the unimodular gauge.
