Generative AI, notably embodied by Large Language Models (LLMs), has the potential to revolutionize key aspects of Business Process Management (BPM). Addressing the inadequacies of general-purpose LLMs in complex BPM tasks, we follow the Design Science Research paradigm and propose ProcessLLM, a specialized LLM fine-tuned to support BPM tasks. Evaluated in both artificial and real-life business settings, ProcessLLM demonstrates promising performance and a sophisticated understanding of BPM concepts and the BPMN language. Our research serves as a foundational exploration at the intersection of BPM and LLMs, paving the way for future research in LLM-enhanced BPM. Validated by business experts, our research also demonstrates the real-world utility and applicability of LLMs in BPM, thus setting the stage for their broader adoption in operational BPM settings.