  We report on the realization of a hemispherical optical cavity with a finesse
of F = 13000 sustaining inter-cavity powers of 10 kW, which we operate in a
closed-cycle cryostat vacuum system close to 4 Kelvin. This was designed and
built with an integrated radio-frequency Paul trap, in order to combine optical
and radio-frequency trapping. The cavity provides a power build-up factor of
2250. We describe a number of aspects of the system design and operation,
including low-vibration mounting and locking including thermal effects at high
powers. Thermal self-locking in the high intracavity power regime was observed
to enhance the passive stability below 1 kHz. Observations made over repeated
cool-downs over a course of a year show a repeatable shift between the ion trap
center and the cavity mode.
