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We report a fiber-optic sensor for detecting low concentrations of hydrogen. The sensor is based on the absorption change of the evanescent fields in a Pd-coated, single-mode tapered fiber. We fabricated a collection of polarization-sensitive and polarization-insensitive devices and we studied the effect of the different parameters on the sensitivity of the device. We found that the sensor's response is wavelength dependent and it can be adjusted with the taper diameter. Dramatic transmission changes were observed when the sensors were exposed to hydrogen concentrations below 4%. We complement the experimental results with theoretical calculations of the attenuation constant of the fundamental fiber-mode.