Thin film nanoscale elements with a curling magnetic structure (vortex) are a promising candidate for future nonvolatile data storage devices. Their properties are strongly influenced by the spin structure in the vortex core. We have used spin-polarized scanning tunneling microscopy on nanoscale iron islands to probe for the first time the internal spin structure of magnetic vortex cores. Using tips coated with a layer of antiferromagnetic chromium, we obtained images of the curling in-plane magnetization around and of the out-of-plane magnetization inside the core region. The experimental data are compared with micromagnetic simulations. The results confirm theoretical predictions that the size and the shape of the vortex core as well as its magnetic field dependence are governed by only two material parameters, the exchange stiffness and the saturation magnetization that determines the stray field energy.