Human language is expressive because it is compositional: the meaning of a sentence (semantics) can be inferred from its structure (syntax). It is commonly believed that language syntax and semantics are processed by distinct brain regions. Here we revisit this claim using precision fMRI methods to capture separation or overlap of function in the brains of individual participants. Contrary to prior claims, we find distributed sensitivity to both syntax and semantics throughout a broad frontotemporal brain network. Our results join a growing body of evidence for an integrated network for language in the human brain within which internal specialization is primarily a matter of degree rather than kind, in contrast with influential proposals that advocate distinct specialization of different brain areas for different types of linguistic functions.