We tested healthy preterm (born near 28 +/- 2 weeks of gestational age) and full-term infants at various different ages. We compared the two populations on the development of a language acquisition landmark, namely, the ability to distinguish the native language from a rhythmically similar one. This ability is attained 4 months after birth in healthy full-term infants. We measured the induced gamma-band power associated with passive listening to (i) the infants' native language (Spanish), (ii) a rhythmically close language (Italian), and (iii) a rhythmically distant language (Japanese) as a marker of gains in language discrimination. Preterm and full-term infants were matched for neural maturation and duration of exposure to broadcast speech. We found that both full-term and preterm infants only display a response to native speech near 6 months after their term age. Neural maturation seems to constrain advances in speech discrimination at early stages of language acquisition.