Biconical tapered single-mode fiber, which is common in many telecommunications components, offers an alternative sensor to typical optical fiber strain gauges that are susceptible to temperature and pressure effects and require expensive and sophisticated signal acquisition systems. Cavity ringdown spectroscopy, a technique commonly applied to high-sensitivity chemical analysis, offers detection sensitivity advantages that can be used to improve strain measurement with biconical tapers. Combining these two technologies in a spatially extended resonator, we demonstrate a minimum detectable change in ringdown time of 0.08%, corresponding to a minimum detectable displacement of 4.8 nm, and a sensitivity to strain as small as 79 n epsilon/square root(Hz) over a 5-mm taper length.