A central problem in language acquisition is how children effortlessly acquire the grammar of their native language even though speech provides no direct information about underlying structure. This learning problem is even more challenging for dual language learners, yet bilingual infants master their mother tongues as efficiently as monolinguals do. Here we ask how bilingual infants succeed, investigating the particularly challenging task of learning two languages with conflicting word orders (English: eat an apple versus Japanese: ringo-wo taberu 'apple.acc eat'). We show that 7-month-old bilinguals use the characteristic prosodic cues (pitch and duration) associated with different word orders to solve this problem. Thus, the complexity of bilingual acquisition is countered by bilinguals' ability to exploit relevant cues. Moreover, the finding that perceptually available cues like prosody can bootstrap grammatical structure adds to our understanding of how and why infants acquire grammar so early and effortlessly.