The spectrum of eigenmodes of a small disk-shaped ferromagnetic particle in the vortex state is studied for weakly anisotropic easy-plane ferromagnet. An exact analytical solution is constructed in the most interesting case of the modes with the lowest frequencies; such eigenfrequencies can be anomalously low in comparison with particles in homogeneous magnetization. For the rest modes the qualitative analysis by the vortex-scattering problem and numerical methods is used; the splitting of doublets with opposite azimuthal quantum numbers m is predicted.