Fine-tuning language models is commonly believed to inevitably harm their safety, i.e., refusing to respond to harmful user requests, even w
… (voir plus)hen using harmless datasets, thus requiring additional safety measures. We challenge this belief through systematic testing, showing that poor optimization choices, rather than inherent trade-offs, often cause safety problems, measured as harmful responses to adversarial prompts. By properly selecting key training hyper-parameters, e.g., learning rate, batch size, and gradient steps, we reduce unsafe model responses from 16\% to approximately 5\%, as measured by keyword matching, while maintaining utility performance. Based on this observation, we propose a simple exponential moving average (EMA) momentum technique in parameter space that preserves safety performance by creating a stable optimization path and retains the original pre-trained model's safety properties. Our experiments on the Llama families across multiple datasets (Dolly, Alpaca, ORCA) demonstrate that safety problems during fine-tuning can largely be avoided without specialized interventions, outperforming existing approaches that require additional safety data while offering practical guidelines for maintaining both model performance and safety during adaptation.