The development of human-robot collaboration has the ability to improve manufacturing system performance by leveraging the unique strengths of both humans and robots. On the shop floor, human operators contribute with their adaptability and flexibility in dynamic situations, while robots provide precision and the ability to perform repetitive tasks. However, the communication gap between human operators and robots limits the collaboration and coordination of human-robot teams in manufacturing systems. Our research presents a human-robot collaborative assembly framework that utilizes a large language model for enhancing communication in manufacturing environments. The framework facilitates human-robot communication by integrating voice commands through natural language for task management. A case study for an assembly task demonstrates the framework's ability to process natural language inputs and address real-time assembly challenges, emphasizing adaptability to language variation and efficiency in error resolution. The results suggest that large language models have the potential to improve human-robot interaction for collaborative manufacturing assembly applications.