Machine Translation (MT) for Ancient Greek (AG) to Modern Greek (MG) is a low-resource task, constrained by the lack of large-scale, high-quality parallel data. We address this gap by introducing the AG-MG Parallel Corpus, a new resource containing 132,481 sentence-aligned pairs derived from literary, historical, and biblical texts. We present a novel corpus creation pipeline that combines web-scraped, excerpt-level data with a multi-stage sentence-level alignment, and refinement process. Our method uses VecAlign with LaBSE embeddings, which we first fine-tune on a manually-aligned AG-MG subset, followed by an LLM-based error/misalignment correction phase using Gemini 2.5 Flash to ensure high alignment quality. Furthermore, we provide the first comprehensive benchmark of modern MT models on this task, evaluating three fine-tuning strategies across NMT models (NLLB, M2M100) and a Greek LLM (Llama-Krikri-8B). Our experiments show that fine-tuning yields significant improvements over base models, increasing performance by up to +10.3 BLEU points. Specifically, full-parameter fine-tuning of Llama-Krikri-8B achieves the highest overall performance with a BLEU score of 13.16, while the QLoRA-adapted M2M100-1.2B model demonstrates the largest relative gains and highly competitive results. Our dataset and models represent a significant contribution to Greek NLP.