Recent advancements in large language models (LLMs) have highlighted the risk of misusing them, raising the need for accurate detection of LLM-generated content. In response, a viable solution is to inject imperceptible identifiers into LLMs, known as watermarks. Our research extends the existing watermarking methods by proposing the novel Sampling One Then Accepting (STA-1) method. STA-1 is an unbiased watermark that preserves the original token distribution in expectation and has a lower risk of producing unsatisfactory outputs in low-entropy scenarios compared to existing unbiased watermarks. In watermark detection, STA-1 does not require prompts or a white-box LLM, provides statistical guarantees, demonstrates high efficiency in detection time, and remains robust against various watermarking attacks. Experimental results on low-entropy and high-entropy datasets demonstrate that STA-1 achieves the above properties simultaneously, making it a desirable solution for watermarking LLMs. Implementation codes for this study are available online.