We report the theoretical description and the experimental demonstration of an optical resonator formed by inserting a Fiber Bragg Grating (FBG) in a closed fiber loop. The spectral characteristics of such a resonator strongly depend on the reflectivity of the FBG. In the wavelength region where the FBG reflectivity R is negligible, the system behaves like a conventional ring resonator. On the other hand, when R is not vanishing, a split-mode structure can be observed, associated to the degeneracy removal of two counterpropagating resonant modes. The magnitude of the mode splitting can be used to sense small variations of the FBG physical parameters, such as length, temperature or group index. An example of strain sensing with this setup is reported, showing that the mode splitting is sensitive to a mechanical strain applied to the FBG, while it is almost insensitive to a strain applied to any other point of the resonator. This peculiar feature allows to perform cavity-enhanced, local strain measurements with a reduced sensitivity to environmental perturbations, which represents an important improvement in the framework of the fiber-optic sensors.