The advancement in spiking neural networks (SNNs) provides a promising and alternative approach to conventional artificial neural networks (ANNs) with higher energy efficiency. However, the significant requirements on memory usage presents a performance bottleneck on resource constrained devices. Inspired by the notion of binarized neural networks (BNNs), we incorporate the design principles in BNNs into that of SNNs to reduce the stringent resource requirements. Specifically, the weights are binarized to 1 and -1 for implementing the functions of excitatory and inhibitory synapses. Hence, the proposed design is referred to as a weight-binarized spiking neural network (WB-SNN). In the WB-SNN, only one bit is used for the weight or a spike; for the latter, 1 and 0 indicate a spike and no spike, respectively. A priority encoder is used to identify the index of an active neuron as a basic unit to construct the WB-SNN. We further design a fully connected neural network that consists of an input layer, an output layer, and fully connected layers of different sizes. A counter is utilized in each neuron to complete the accumulation of weights. The WB-SNN design is validated by using a multi-layer perceptron on the MNIST dataset. Hardware implementations on FPGAs show that the WB-SNN attains a significant saving of memory with only a limited accuracy loss compared with its SNN and BNN counterparts.