A finesse of 21 has been measured for a fiber Fabry-Perot interferometer (FFPI) with multilayer TiO2/SiO2 internal mirrors. Experimental reflectance and transmittance data for the thermally tuned interferometer were fit with calculated curves using values of 86% for the reflectance and 7.2% (0.33 dB) for the excess optical loss of each mirror. Applications in sensing and communications are discussed.