We derive a closed-form solution for the Green's function for the wave equation of a static (with respect to an undragged, static observer at infinity) scalar charge in the Kerr space-time. We employ our solution to obtain an analytic expression for the self-force on such a charge, comparing our results to those of L. M. Burko and Y. T. Liu, Phys. Rev. D 64, 024006 (2001).