Neuromorphic computing systems are inspired by humans brains, where data are stored and processed at the same location. Contrary to von Neumann systems, neuromorphic computing systems offer excellent real-time processing for huge data sizes, at low costs and power consumption. Most of these systems rely on emerging new devices, such as memristors, to build crossbar arrays implementing different neural network topologies. The Echo State Network model is a special type of recurrent neural networks, which can correctly represent spatiotemporal dataset. In this paper, a new hardware implementation design for the Echo State Network model using memristor double crossbar arrays is proposed. Moreover, a detailed design procedure is proposed for designing and simulating the proposed architecture. The system has been evaluated using the ubiquitous Mackey-Glass dataset showing promising results, compared to the software implementation of the model. In addition, the system shows excellent immunity against memristor process variations.