In this study, we explore the effectiveness of fine-tuning Large Language Models (LLMs), particularly Llama 3 8B Instruct, using translation memories (TMs) for hyper-specific machine translation (MT) tasks. Decoder-only LLMs have shown impressive performance in MT due to their ability to learn from extensive datasets and generate high quality translations. However, LLMs often struggle with the nuances and style required for organisation-specific translation so we leverage TMs, which store human translated segments, as a valuable resource to enhance translation accuracy and efficiency. We investigate the impact of fine-tuning the Llama 3 model using TMs from a specific organisation in the software sector. Our experiments cover five translation directions across languages of varying resource levels (English to Brazilian Portuguese, Czech, German, Finnish, and Korean). We analyse diverse sizes of training datasets (1k to 100k+ segments) to evaluate their influence on translation quality. We fine-tune separate models for each training set and evaluate their performance based on automatic metrics, BLEU, chrF++, TER, and COMET. Our findings reveal improvement in translation performance with larger datasets across all metrics. On average, BLEU and COMET scores increase by 13 and 25 points respectively on the largest training set against the baseline model. Notably, there is a performance deterioration in comparison with the baseline model when fine-tuning on only 1k and 2k examples; however, we observe a substantial improvement as the training dataset size increases. The study highlights the potential of integrating TMs with LLMs to create bespoke translation models tailored to the specific needs of businesses, therefore enhancing translation quality and reducing turn-around times. This approach offers a valuable insight for organisations seeking to leverage TMs and LLMs for optimal translation outcomes, specially in narrower domains.