Necessary and sufficient conditions for the addivity of ordinal utility functions are well known. In this paper a necessary and sufficient condition for the addivity of von Neumann-Morgenstern utility functions is presented; this condition is summarized in the "strong additivity axiom". A "weak addivity axiom" is introduced, and it is shown that an individual's preferences satisfy this axiom if and only if his von Neumann-Morgenstern utility function is either additive or "log additive". The extension of these results to the continuous consumption case is indicated.