  We prove that an integrated version of the Gurau colored tensor model
supplemented with the usual Bosonic propagator on $U(1)^4$ is renormalizable to
all orders in perturbation theory. The model is of the type expected for
quantization of space-time in 4D Euclidean gravity and is the first example of
a renormalizable model of this kind. Its vertex and propagator are
four-stranded like in 4D group field theories, but without gauge averaging on
the strands. Surprisingly perhaps, the model is of the $\phi^6$ rather than of
the $\phi^4$ type, since two different $\phi^6$-type interactions are
log-divergent, i.e. marginal in the renormalization group sense. The
renormalization proof relies on a multiscale analysis. It identifies all
divergent graphs through a power counting theorem. These divergent graphs have
internal and external structure of a particular kind called melonic. Melonic
graphs dominate the 1/N expansion of colored tensor models and generalize the
planar ribbon graphs of matrix models. A new locality principle is established
for this category of graphs which allows to renormalize their divergences
through counterterms of the form of the bare Lagrangian interactions. The model
also has an unexpected anomalous log-divergent $(\int \phi^2)^2$ term, which
can be interpreted as the generation of a scalar matter field out of pure
gravity.
