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We report on the realization of a hemispherical optical cavity with a finesseof F = 13000 sustaining inter-cavity powers of 10 kW, which we operate in aclosed-cycle cryostat vacuum system close to 4 Kelvin. This was designed andbuilt with an integrated radio-frequency Paul trap, in order to combine opticaland radio-frequency trapping. The cavity provides a power build-up factor of2250. We describe a number of aspects of the system design and operation,including low-vibration mounting and locking including thermal effects at highpowers. Thermal self-locking in the high intracavity power regime was observedto enhance the passive stability below 1 kHz. Observations made over repeatedcool-downs over a course of a year show a repeatable shift between the ion trapcenter and the cavity mode.

Integration of a high finesse cryogenic build-up cavity with an ion trap