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Current federated learning algorithms take tens of communication rounds transmitting unwieldy model weights under ideal circumstances and hundreds when data is poorly distributed. Inspired by recent work on dataset distillation and distributed one-shot learning, we propose Distilled One-Shot Federated Learning (DOSFL) to significantly reduce the communication cost while achieving comparable performance. In just one round, each client distills their private dataset, sends the synthetic data (e.g. images or sentences) to the server, and collectively trains a global model. The distilled data look like noise and are only useful to the specific model weights, i.e., become useless after the model updates. With this weight-less and gradient-less design, the total communication cost of DOSFL is up to three orders of magnitude less than FedAvg while preserving between 93% to 99% performance of a centralized counterpart. Afterwards, clients could switch to traditional methods such as FedAvg to finetune the last few percent to fit personalized local models with local datasets. Through comprehensive experiments, we show the accuracy and communication performance of DOSFL on both vision and language tasks with different models including CNN, LSTM, Transformer, etc. We demonstrate that an eavesdropping attacker cannot properly train a good model using the leaked distilled data, without knowing the initial model weights. DOSFL serves as an inexpensive method to quickly converge on a performant pre-trained model with less than 0.1% communication cost of traditional methods.