Language discrimination is one of the core differences between bilingual and monolingual language acquisition. Here, we investigate the earliest brain specialization induced by it. Following previous research, we hypothesize that bilingual native language discrimination is a complex process involving specific processing of the prosodic properties of the speech signal. We recorded the brain activity of monolingual and bilingual 4.5-month-old infants using EEG, while listening to their native/dominant language and two foreign languages. We defined two different windows of analysis to separate discrimination and identification effects. In the early window of analysis (150-280 ms) we measured the P200 component, and in the later window of analysis we measured Theta (400-1800 ms) and Gamma (300-2800 ms) oscillations. The results point in the direction of different language discrimination strategies for bilingual and monolingual infants. While only monolingual infants show early discrimination of their native language based on familiarity, bilinguals perform a later processing which is compatible with an increase in attention to the speech signal. This is the earliest evidence found for brain specialization induced by bilingualism.