The dynamics of propagating polariton condensates in one-dimensional microcavities is investigated through time resolved experiments. We find a strong increase in the condensate intensity when it travels through the nonresonantly excited area. This amplification is shown to come from bosonic stimulated relaxation of reservoir excitons into the polariton condensate, allowing for the repopulation of the condensate through nonresonant pumping. Thus, we experimentally demonstrate a polariton amplifier with a large band width, opening the way towards the transport of polaritons with high densities over macroscopic distances.