A $C^{\infty }$-Hopf algebra is a $C^{\infty }$-algebra which is also a convenient Hopf algebra with respect to the structure induced by the evaluations of smooth functions. We characterize those $C^{\infty }$-Hopf algebras which are given by the algebra $C^{\infty }\left(G\right)$ of smooth functions on some compact Lie group $G$, thus obtaining an anti-isomorphism of the category of compact Lie groups with a subcategory of convenient Hopf algebras.