We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. Th
e techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.
