To investigate whether different language deficits are distinguished by the relative strengths of their association with the functional connectivity (FC) at rest of the language network (LN) and cingulo-opercular network (CON) after aphasic stroke. In a group of patients with acute stroke and left-hemisphere damage, we identified 3 distinct, yet correlated, clusters of deficits including comprehension/lexical semantic, grapheme-phoneme knowledge, and verbal executive functions. We computed partial correlations in which the contributions of a behavioral cluster and network FC of no interest were statistically regressed out. We observed a double dissociation such that impairment of grapheme-phoneme knowledge was more associated with lower FC of the LN within the left hemisphere than lower FC of the CON, whereas verbal executive deficits were more related to lower FC of the CON than the LN in the left hemisphere. Furthermore, the specific association between language deficits and FC was independent of the amount of structural damage to the LN and CON. These findings indicate that after a left-hemisphere lesion, the type of language impairment is related to the abnormal pattern of correlated activity in different networks. Accordingly, they extend the concept of a neuropsychological double dissociation from structural damage to functional network abnormalities. Finally, current results strongly argue in favor of the behavioral specificity of intrinsic brain activity after focal structural damage.