We extend John's inscribed ellipsoid theorem, as well as Loewner's circum-scribed ellipsoid theorem, from convex bodies to proper cones. To be more precise, we prove that a proper cone K in R n contains a unique ellipsoidal cone Q in (K) of maximal canonical volume and, on the other hand, it is enclosed by a unique ellipsoidal cone Q circ (K) of minimal canonical volume. In addition, we explain how to construct the inscribed ellipsoidal cone Q in (K). The circumscribed ellipsoidal cone Q circ (K) is then obtained by duality arguments. The canonical volume of an ellipsoidal cone is defined as the usual n-dimensional volume of a certain truncation of the cone.