A new numerical method for the solution of the homogeneous Boltzmann equation on nonuniform grids is developed. The collision operator is written using the Fourier transform. This formulation and a special new discretization of the gain part allow for the fast numerical computations on nonuniform grids in velocity space. The computational cost of the algorithm is $O(N_v)+O(\sigma N^6 \log N)$ for a general model of interaction. Here $N_v$ is the number of velocity points, $N$ denotes the number of modes in the Fourier domain and $\sigma<1$ is a small constant depending on the discretization. The results of some numerical test are presented.