For bilinguals, speaking in a second language (L2) compared to the native language (L1) is usually more difficult. In this study we asked whether the difficulty in L2 production reflects increased demands imposed on domain-general or core language mechanisms. We compared the brain response to speech production in L1 and L2 within two functionally-defined networks in the brain: the Multiple Demand (MD) network and the language network. We found that speech production in L2 was linked to a widespread increase of brain activity in the domain-general MD network. The language network did not show a similarly robust differences in processing speech in the two languages, however, we found increased response to L2 production in the language-specific portion of the left inferior frontal gyrus (IFG). To further explore our results, we have looked at domain-general and language-specific response within the brain structures postulated to form a Bilingual Language Control (BLC) network. Within this network, we found a robust increase in response to L2 in the domain-general, but also in some language-specific voxels including in the left IFG. Our findings show that L2 production strongly engages domain-general mechanisms, but only affects language sensitive portions of the left IFG. These results put constraints on the current model of bilingual language control by precisely disentangling the domain-general and language-specific contributions to the difficulty in speech production in L2.