Large Language Models (LLMs) have shown remarkable capabilities across various domains, but their potential for solving combinatorial optimization problems remains largely unexplored. In this paper, we investigate the applicability of LLMs to the Job Shop Scheduling Problem (JSSP), a classic challenge in combinatorial optimization that requires efficient job allocation to machines to minimize makespan. To this end, we introduce Starjob, the first supervised dataset for JSSP, comprising 130k instances specifically designed for training LLMs. Leveraging this dataset, we fine-tune the LLaMA 8B 4-bit quantized model with the LoRA method to develop an end-to-end scheduling approach. Our evaluation on standard benchmarks demonstrates that the proposed LLM-based method not only surpasses traditional Priority Dispatching Rules (PDRs) but also achieves notable improvements over state-of-the-art neural approaches like L2D, with an average improvement of 15.36% on DMU and 7.85% on Taillard benchmarks. These results highlight the untapped potential of LLMs in tackling combinatorial optimization problems, paving the way for future advancements in this area.