Recent advances in large language models (LLMs) show potential for clinical applications, such as clinical decision support and trial recommendations. However, the GPT-4 LLM predicts an excessive number of ICD codes for medical coding tasks, leading to high recall but low precision. To tackle this challenge, we introduce LLM-codex, a two-stage approach to predict ICD codes that first generates evidence proposals using an LLM and then employs an LSTM-based verification stage. The LSTM learns from both the LLM's high recall and human expert's high precision, using a custom loss function. Our model is the only approach that simultaneously achieves state-of-the-art results in medical coding accuracy, accuracy on rare codes, and sentence-level evidence identification to support coding decisions without training on human-annotated evidence according to experiments on the MIMIC dataset.