We consider a dual pair (G, G'), in the sense of Howe, with G compact acting on L^2(R^n), for an appropriate n, via the Weil representation $\omega$. Let \wt\G be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation \Pi of \wt\G, let \Pi' be the corresponding irreducible unitary representation of \wt\G' in the Howe duality. The orthogonal projection onto L^2(\R^n)_\Pi, the \Pi-isotypic component, is the essentially unique symmetry breaking operator in Hom_{\wt\G \wt\G'}(\Hc_\omega^\infty, \Hc_\Pi^\infty\otimes \Hc_{\Pi'}^\infty). We study this operator by computing its Weyl symbol. Our results allow us to compute the wavefront set of \Pi' by elementary means.