  We investigate the spectral properties of self-adjoint Schr\"odinger
operators with attractive $\delta$-interactions of constant strength $\alpha >
0$ supported on conical surfaces in ${\mathbb R}^3$. It is shown that the
essential spectrum is given by $[-\alpha^2/4,+\infty)$ and that the discrete
spectrum is infinite and accumulates to $-\alpha^2/4$. Furthermore, an
asymptotic estimate of these eigenvalues is obtained.
