In this paper, we propose to improve the performance of impulse-radio ultra-wideband (IR-UWB) systems by means of cooperative relaying. With regard to a simple practical realization, we focus on a non-coherent system setup in conjunction with amplify-and-forward (A & F) relaying. In particular, considering a two-hop scenario we propose to employ a double-differential encoding scheme at the source node and single differential decoding at the relay and the destination node, respectively, so as to efficiently limit intersymbol-interference (ISI) effects. A thorough performance analysis of the proposed scheme is provided, along with a closed-form optimization of the transmit power allocation between source and relay. Simulation results illustrate the excellent performance of the proposed scheme, which is compared to alternative coherent and non-coherent schemes based on A & F relaying and decode-and-forward (D & F) relaying.