  In dimension greater than or equal to three, we investigate the spectrum of a
Schr{\"o}dinger operator with a $\delta$-interaction supported on a cone whose
cross section is the sphere of co-dimension two. After decomposing into fibers,
we prove that there is discrete spectrum only in dimension three and that it is
generated by the axisymmetric fiber. We get that these eigenvalues are
non-decreasing functions of the aperture of the cone and we exhibit the precise
logarithmic accumulation of the discrete spectrum below the threshold of the
essential spectrum.
