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We demonstrate a fiber Fabry Pérot cavity in the ultraviolet range,which covers the florescence wavelength for the 2P1/2 to 2S1/2 transition of Yb+ and is designed in the bad cavity limit for florescence collection.Benefiting from both the small cavity mode volume and the large atom dipole,a cavity with moderate finesse and high transmission still supports a good cooperativity,which is made and tested in experiment.Based on the measured experimental parameters,simulation performed on the cavity and ion shows a Purcell factor better than 2.5 and a single-mode fiber collection efficiency over 10％.This technology can support ultra-bright single photon sources based on trapped ions and can provide the possibility to link remote atoms as a quantum network.