The "syntactic bootstrapping" hypothesis proposes that syntactic structure provides children with cues for learning the meaning of novel words. In this article, we address the question of how children might start acquiring some aspects of syntax before they possess a sizeable lexicon. The study presents two models of early syntax acquisition that rest on three major assumptions grounded in the infant literature: First, infants have access to phrasal prosody; second, they pay attention to words situated at the edges of prosodic boundaries; third, they know the meaning of a handful of words. The models take as input a corpus of French child-directed speech tagged with prosodic boundaries and assign syntactic labels to prosodic phrases. The excellent performance of these models shows the feasibility of the syntactic bootstrapping hypothesis, since elements of syntactic structure can be constructed by relying on prosody, function words, and a minimal semantic knowledge.