Title: Data-Free One-Shot Federated Learning Under Very High Statistical Heterogeneity. Abstract: Federated learning (FL) is an emerging distributed learning framework that collaboratively trains a shared model without transferring the local clients' data to a centralized server. Motivated by concerns stemming from extended communication and potential attacks, one-shot FL limits communication to a single round while attempting to retain performance. However, one-shot FL methods often degrade under high statistical heterogeneity, fail to promote pipeline security, or require an auxiliary public dataset. To address these limitations, we propose two novel data-free one-shot FL methods: FedCVAE-Ens and its extension FedCVAE-KD. Both approaches reframe the local learning task using a conditional variational autoencoder (CVAE) to address high statistical heterogeneity. Furthermore, FedCVAE-KD leverages knowledge distillation to compress the ensemble of client decoders into a single decoder. We propose a method that shifts the center of the CVAE prior distribution and experimentally demonstrate that this promotes security, and show how either method can incorporate heterogeneous local models. We confirm the efficacy of the proposed methods over baselines under high statistical heterogeneity using multiple benchmark datasets. In particular, at the highest levels of statistical heterogeneity, both FedCVAE-Ens and FedCVAE-KD typically more than double the accuracy of the baselines.