We analyze a new formulation of the Stokes equations inthree-dimensional axisymmetric geometries, relying on Fourier expansion with respect tothe angular variable: the problem for each Fourier coefficient is two-dimensional and hassix scalar unknowns, corresponding to the vector potential and the vorticity. Aspectral discretization is built on this formulation, which leads to an exactlydivergence-free discrete velocity. We prove optimal error estimates.