People learn new languages with varying degrees of success but what are the neuroanatomical correlates of the difference in language-learning aptitude? In this study, we set out to investigate how differences in cortical morphology and white matter microstructure correlate with aptitudes for vocabulary learning, phonetic memory, and grammatical inferencing as measured by the first-language neutral LLAMA test battery. We used ultra-high field (7T) magnetic resonance imaging to estimate the cortical thickness and surface area from sub-millimeter resolved image volumes. Further, diffusion kurtosis imaging was used to map diffusion properties related to the tissue microstructure from known language-related white matter tracts. We found a correlation between cortical surface area in the left posterior-inferior precuneus and vocabulary learning aptitude, possibly indicating a greater predisposition for storing word-figure associations. Moreover, we report negative correlations between scores for phonetic memory and axial kurtosis in left arcuate fasciculus as well as mean kurtosis, axial kurtosis, and radial kurtosis of the left superior longitudinal fasciculus III, which are tracts connecting cortical areas important for phonological working memory.