The paper deals with the following question: Given a proper convex functionfonRnsuch that domfj≠Rn, does there exist a convex function finite on all ofRnwhich agrees withfon domf? A necessary and sufficient condition for the existence of such a function is derived. Moreover, the existence of a .minimal function in the collection of all these functions is proved in the case where domfhas a nonempty interior. Examples illuminating some of the difficulties arising are considered at the beginning of the paper.