We present experimental and theoretical studies of polarization-gradient cooling of metastable neon atoms confined to the dark center of a TEM*01 (doughnut) mode. A slow beam of neon atoms is guided and focused inside a blue-detuned and focused doughnut-mode laser beam to a spot size below 10 μm. The transverse motion inside this doughnut mode is cooled by means of two-dimensional optical molasses. We observed non-Gaussian two-component velocity distributions of which the cold component has a width of down to three recoil velocities. These results are found to be in qualitative agreement with a quantum Monte Carlo simulation of cooling in one dimension in the presence of an external light-shift potential. For the simulation we apply the recently developed technique of quantum-state diffusion with adaptive noise.