As large language models (LLMs) are increasingly used for factual question-answering, it becomes more important for LLMs to have the capability to communicate the likelihood that their answer is correct. For these verbalized expressions of uncertainty to be meaningful, they should reflect the error rates at the expressed level of confidence. However, when prompted to express confidence, the error rates of current LLMs are inconsistent with their communicated confidences, highlighting the need for uncertainty quantification methods. Many prior methods calculate lexical uncertainty, estimating a model's confidence in the specific string it generated. In some cases, however, it may be more useful to estimate semantic uncertainty, or the model's confidence in the answer regardless of how it is verbalized. We propose a simple procedure, uncertainty distillation, to teach an LLM to verbalize calibrated semantic confidences. Using held-out data to map initial uncertainty estimates to meaningful probabilities, we create examples annotated with verbalized probabilities for supervised fine-tuning. We find that our method yields verbalized confidences that correlate well with observed error rates, even when compared to strong baselines, some of which are more than twenty times slower at inference time. Additionally, we demonstrate that our method can be applied to black-box models that allow API-based fine-tuning, resulting in estimates of uncertainty that are both more effective and more efficient than any of our baselines.