Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if H acts on A \\'inner-faithfully\\', then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen. This result also extends to working over k of positive characteristic. We also discuss results on Hopf actions on Weyl algebras as a consequence of the main theorem. This is joint work with Pavel Etingof.