The classical motion at a conical surface is bounded at one (narrower) side
of the cone and unbounded at the other. However, it is shown here that a
dielectric cone with a small half-angle gamma can perform as a high Q-factor
optical microresonator which completely confines light. The theory of the
discovered localized conical states is in excellent agreement with experimental
data. It provides both a unique approach for extremely accurate local
characterization of optical fibers (which usually have gamma ~10^-5 or less)
and a new paradigm in the field of high Q-factor resonators