We present an automatic large language model (LLM) conversion approach that produces uncertainty-aware LLMs capable of estimating uncertainty with every prediction. Our approach is model- and data-agnostic, is computationally-efficient, and does not rely on external models or systems. We evaluate converted models on the selective question answering setting -- to answer as many questions as possible while maintaining a given accuracy, forgoing providing predictions when necessary. As part of our results, we test BERT and Llama 2 model variants on the SQuAD extractive QA task and the TruthfulQA generative QA task. We show that using the uncertainty estimates provided by our approach to selectively answer questions leads to significantly higher accuracy over directly using model probabilities.