Large Language Models (LLMs) have shown impressive performance across various domains, but their ability to perform molecular reasoning remains underexplored. Existing methods mostly rely on general-purpose prompting, which lacks domain-specific molecular semantics, or fine-tuning, which faces challenges in interpretability and reasoning depth, often leading to structural and textual hallucinations. To address these issues, we introduce MolReasoner, a two-stage framework that transitions LLMs from memorization to high-fidelity chemical reasoning. In the Mol-SFT stage, knowledge-enhanced Chain-of-Thought (CoT) data provides a strong foundation, while the Mol-RL stage refines reasoning using a novel, task-adaptive reward system to mitigate hallucinations. Extensive evaluations demonstrate that MolReasoner significantly outperforms a wide range of strong baselines in both molecule generation and captioning tasks. Further analyses highlight the framework's synergistic design and its ability to produce more interpretable outputs. Our work presents a principled and effective new approach for advancing high-fidelity molecular reasoning.