  In the two previous papers of this series we defined a new combinatorical
approach to quantum gravity, Algebraic Quantum Gravity (AQG). We showed that
AQG reproduces the correct infinitesimal dynamics in the semiclassical limit,
provided one incorrectly substitutes the non -- Abelean group SU(2) by the
Abelean group $U(1)^3$ in the calculations. The mere reason why that
substitution was performed at all is that in the non -- Abelean case the volume
operator, pivotal for the definition of the dynamics, is not diagonisable by
analytical methods. This, in contrast to the Abelean case, so far prohibited
semiclassical computations. In this paper we show why this unjustified
substitution nevertheless reproduces the correct physical result: Namely, we
introduce for the first time semiclassical perturbation theory within AQG (and
LQG) which allows to compute expectation values of interesting operators such
as the master constraint as a power series in $\hbar$ with error control. That
is, in particular matrix elements of fractional powers of the volume operator
can be computed with extremely high precision for sufficiently large power of
$\hbar$ in the $\hbar$ expansion. With this new tool, the non -- Abelean
calculation, although technically more involved, is then exactly analogous to
the Abelean calculation, thus justifying the Abelean analysis in retrospect.
The results of this paper turn AQG into a calculational discipline.
