Spiking neural networks (SNNs) can be run on neuromorphic devices with ultra-high speed and ultra-low energy consumption because of their binary and event-driven nature. Therefore, SNNs are expected to have various applications, including as generative models being running on edge devices to create high-quality images. In this study, we build a variational autoencoder (VAE) with SNN to enable image generation. VAE is known for its stability among generative models; recently, its quality advanced. In vanilla VAE, the latent space is represented as a normal distribution, and floating-point calculations are required in sampling. However, this is not possible in SNNs because all features must be binary time series data. Therefore, we constructed the latent space with an autoregressive SNN model, and randomly selected samples from its output to sample the latent variables. This allows the latent variables to follow the Bernoulli process and allows variational learning. Thus, we build the Fully Spiking Variational Autoencoder where all modules are constructed with SNN. To the best of our knowledge, we are the first to build a VAE only with SNN layers. We experimented with several datasets, and confirmed that it can generate images with the same or better quality compared to conventional ANNs. The code is available at https://github.com/kamata1729/FullySpikingVAE.