The application of machine learning on tabular data in specialized domains is severely limited by data scarcity. While generative models offer a solution, traditional methods falter in low-data regimes, and recent Large Language Models (LLMs) often ignore the explicit dependency structure of tabular data, leading to low-fidelity synthetics. To address these limitations, we introduce StructSynth, a novel framework that integrates the generative power of LLMs with robust structural control. StructSynth employs a two-stage architecture. First, it performs explicit structure discovery to learn a Directed Acyclic Graph (DAG) from the available data. Second, this learned structure serves as a high-fidelity blueprint to steer the LLM's generation process, forcing it to adhere to the learned feature dependencies and thereby ensuring the generated data respects the underlying structure by design. Our extensive experiments demonstrate that StructSynth produces synthetic data with significantly higher structural integrity and downstream utility than state-of-the-art methods. It proves especially effective in challenging low-data scenarios, successfully navigating the trade-off between privacy preservation and statistical fidelity.