This paper studies a three-node network in which an intermediate nano-transceiver, acting as a relay, is placed between a nano-transmitter and a nano-receiver to improve the range of diffusion-based molecular communication. Motivated by the relaying protocols used in traditional wireless communication systems, we study amplify-and-forward (AF) relaying with fixed and variable amplification factor for use in molecular communication systems. To this end, we derive a closed-form expression for the expected end-to-end error probability. Furthermore, we derive a closed-form expression for the optimal amplification factor at the relay node for minimization of an approximation of the expected error probability of the network. Our analytical and simulation results show the potential of AF relaying to improve the overall performance of nano-networks.