A novel measurement principle for fiber-optic sensing is presented. Use of a cavity-ring-down scheme enables measurements of minute optical losses in high-finesse fiber-optic cavities. The loss may be induced by evanescent-field absorption, fiber bending, fiber degradation, Bragg gratings, or any other effect that might change the fiber transmission or cavity reflector properties. The principle is proved to be rather insensitive to ambient perturbations such as temperature changes. A high-sensitivity measurement of loss due to bending is presented as a proof-of-principle. With a cavity finesse of 627 a sensitivity for induced loss of 108 ppm (4.68 x 10(-4) dB) is achieved. Preliminary measurements of evanescent-field absorption are also discussed.