The efficacy of an large language model (LLM) generated text detector depends substantially on the availability of sizable training data. White-box zero-shot detectors, which require no such data, are nonetheless limited by the accessibility of the source model of the LLM-generated text. In this paper, we propose an simple but effective black-box zero-shot detection approach, predicated on the observation that human-written texts typically contain more grammatical errors than LLM-generated texts. This approach entails computing the Grammar Error Correction Score (GECScore) for the given text to distinguish between human-written and LLM-generated text. Extensive experimental results show that our method outperforms current state-of-the-art (SOTA) zero-shot and supervised methods, achieving an average AUROC of 98.7% and showing strong robustness against paraphrase and adversarial perturbation attacks.