The core structure of a disclination line of strength 1 in uniaxial nematics is determined by using a space-dependent mean-field approach somehow based on the rod-like molecular model. It is shown that, due to distortion, biaxiality arises at all points of the defect core, except at the centre line where symmetry dictates uniaxiality; the orientational distribution of the rod-like molecules is there 'oblate', however. Although, as is well known, the configuration of a 1 defect is stable only under limiting conditions (as compared with either escaping in the axial direction or splitting into two 1/2 lines), the simple example developed here is indicative of a method which can readily be extended to more realistic, if mathematically more complex, situations such as 1/2 lines, layers close to solid boundaries, etc.