When discretizing the Navier-Stokes equations with an IMEX scheme in time, the modified Stokes equation must be solved to advance the solution from one time step to the next. The Green's function for the modified Stokes equation is given in terms of the difference of the Green's functions for the Laplace and modified Helmholtz equations. Unfortunately, it is unstable, particularly on fine grids or at low Reynolds number, to evaluate the Green's function by first evaluating the Laplace and modified Helmholtz Green's functions and then taking the difference. As an alternative, we present a fast multipole method for the modified Stokes kernel itself, which leverages new special functions to stably evaluate the interactions.