In melt-blown fibers there exists a radial variation in molecular weight. This difference is caused by the very rapid (<25 ms) diffusion and thermoxidation reaction of oxygen with the polymer in the fiber sheath. The core remains unaffected. The difference in molecular weight makes the fibers suitable for thermal bonding. A technique based on intrinsic viscosity measurements was developed for measuring sheath and core molecular weights. A complex mathematical model was developed to account for rapid diffusion and thermoxidation. The experimental molecular weight measurements are well-fit by the model.