An end-to-end argument mining (AM) pipeline takes a text as input and provides its argumentative structure as output by identifying and classifying the argument units and argument relations in the text. In this work, we approach AM using fine-tuned large language models (LLMs). We model the three main sub-tasks of the AM pipeline, as well as their joint formulation, as text generation tasks. We fine-tune eight popular quantized and non-quantized LLMs – LLaMA-3, LLaMA-3.1, Gemma-2, Mistral, Phi-3, Qwen-2 – which are among the most capable open-weight models, on the benchmark PE, AbstRCT, and CDCP datasets that represent diverse data sources. Our approach achieves state-of-the-art results across all AM sub-tasks and datasets, showing significant improvements over previous benchmarks.