Reasoning LLMs have demonstrated remarkable breakthroughs in solving complex problems that were previously out of reach. To ensure LLMs do not assist with harmful requests, safety alignment fine-tuning is necessary in the post-training phase. However, safety alignment fine-tuning has recently been shown to significantly degrade reasoning abilities, a phenomenon known as the "Safety Tax". In this work, we show that using LoRA for SFT on refusal datasets effectively aligns the model for safety without harming its reasoning capabilities. This is because restricting the safety weight updates to a low-rank space minimizes the interference with the reasoning weights. Our extensive experiments across four benchmarks covering math, science, and coding show that this approach produces highly safe LLMs -- with safety levels comparable to full-model fine-tuning -- without compromising their reasoning abilities. Additionally, we observe that LoRA induces weight updates with smaller overlap with the initial weights compared to full-model fine-tuning. We also explore methods that further reduce such overlap -- via regularization or during weight merging -- and observe some improvement on certain tasks. We hope this result motivates designing approaches that yield more consistent improvements in the reasoning-safety trade-off.