We demonstrate cooling of the motion of a single neutral atom confined by a dipole trap inside a high-finesse optical resonator. Cooling of the vibrational motion results from EIT-like interference in an atomic Λ-type configuration, where one transition is strongly coupled to the cavity mode and the other is driven by an external control laser. Good qualitative agreement with the theoretical predictions is found for the explored parameter ranges. Further we demonstrate EIT-cooling of atoms in the dipole trap in free space, reaching the ground state of axial motion. By means of a direct comparison with the cooling inside the resonator, the role of the cavity becomes evident by an additional cooling resonance. These results pave the way towards a controlled interaction between atomic, photonic and mechanical degrees of freedom.