We present a proof-of-concept experiment aimed at increasing the sensitivity of Fiber-Bragg-gratings temperature sensors by making use of a weak-value-amplification scheme. The technique requires only linear optics elements for its implementation and appears as a promising method for increasing the sensitivity than state-of the-art sensors can currently provide. The device implemented here is able to generate a shift of the centroid of the spectrum of a pulse of ∼0.035 nm/°C, a nearly fourfold increase in sensitivity over the same fiber-Bragg-grating system interrogated using standard methods.