The measurement of relative displacements and deformations is important in many fields such as structural engineering, aerospace, geophysics, and nanotechnology. Optical-fiber sensors have become key tools for strain measurements, with sensitivity limits ranging between 10(-9) and 10(-6)epsilon hertz (Hz)(-1/2) (where epsilon is the fractional length change). We report on strain measurements at the 10(-13)epsilon-Hz(-1/2) level using a fiber Bragg-grating resonator with a diode-laser source that is stabilized against a quartz-disciplined optical frequency comb, thus approaching detection limits set by thermodynamic phase fluctuations in the fiber. This scheme may provide a route to a new generation of strain sensors that is entirely based on fiber-optic systems, which are aimed at measuring fundamental physical quantities; for example, in gyroscopes, accelerometers, and gravity experiments.