Syntax, the abstract structure of language, is a hallmark of human cognition. Despite its importance, its neural underpinnings remain obscured by inherent limitations of non-invasive brain measures and a near total focus on comprehension paradigms. Here, we address these limitations with high-resolution neurosurgical recordings (electrocorticography) and a controlled sentence production experiment. We uncover three syntactic networks that are broadly distributed across traditional language regions, but with focal concentrations in middle and inferior frontal gyri. In contrast to previous findings from comprehension studies, these networks process syntax mostly to the exclusion of words and meaning, supporting a cognitive architecture with a distinct syntactic system. Most strikingly, our data reveal an unexpected property of syntax: it is encoded independent of neural activity levels. We propose that this “low-activity coding” scheme represents a novel mechanism for encoding information, reserved for higher-order cognition more broadly.