  We show that Einstein's conformal gravity is able to explain simply on the
geometric ground the galactic rotation curves without need to introduce any
modification in both the gravitational as well as in the matter sector of the
theory. The geometry of each galaxy is described by a metric obtained making a
singular rescaling of the Schwarzschild's spacetime. The new exact solution,
which is asymptotically Anti-de Sitter, manifests an unattainable singularity
at infinity that can not be reached in finite proper time, namely, the
spacetime is geodetically complete. It deserves to be notice that we here think
different from the usual. Indeed, instead of making the metric
singularity-free, we make it apparently but harmlessly even more singular then
the Schwarzschild's one. Finally, it is crucial to point that the Weyl's
conformal symmetry is spontaneously broken to the new singular vacuum rather
then the asymptotically flat Schwarzschild's one. The metric, is unique
according to: the null energy condition, the zero acceleration for photons in
the Newtonian regime, and the homogeneity of the Universe at large scales. Once
the matter is conformally coupled to gravity, the orbital velocity for a probe
star in the galaxy turns out to be asymptotically constant consistently with
the observations and the Tully-Fisher relation. Therefore, we compare our model
with a sample of 175 galaxies and we show that our velocity profile very well
interpolates the galactic rotation-curves for a proper choice of the only free
parameter in the metric and the the mass to luminosity ratios, which turn out
to be close to 1 consistently with the absence of dark matter.
