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We report on the fabrication and characterization of the first accelerometer based on a polymer optical fiber Bragg grating (FBG) for operation at both 850 and 1550 nm. The devices have a flat frequency response over a 1-kHz bandwidth and a resonance frequency of about 3 kHz. The response is linear up to at least 15 g and sensitivities as high as 19 pm/g (shift in resonance wavelength per unit acceleration) have been demonstrated. Given that 15 g corresponds to a strain of less than 0.02% and that polymer fibers have an elastic limit of more than 1%, the polymer FBG accelerometer can measure very strong accelerations. We compare with corresponding silica FBG accelerometers and demonstrate that using polymer FBGs improves the sensitivity by more than a factor of four and increases the figure of merit, defined as the sensitivity times the resonance frequency squared.