The structure of micelles of an asymmetric polystyrene-polyisoprene (PS-PT) diblock copolymer has been studied in di-n-butyl phthalate (DBP), which is a slightly selective solvent for PS. The polymer has a molecular weight of 11 500 for the fully deuterated d-PS block and 48 500 for the PI block. Contrast variation SANS using mixtures of protonated and deuterated solvent and a polymer volume fraction of 18% at 10 degrees C were performed to get detailed information on the structure of the micelles. An additional contrast was obtained by performing a small-angle X-ray scattering measurement on a sample in protonated DBP. It was found that micelles are formed with an anisometric PI-rich core surrounded by dissolved d-PS chains. The data were analyzed using model form factors of micelles with, respectively, cylindrical and ellipsoidal cores. It was necessary to include a term to describe the concentration fluctuations within the swollen PI cores. The particle interference effects originating from the high volume fraction were described by a hard-sphere structure factor. Both models fit the data well, however, a cylindrical model which includes size polydispersity gives the best fit.