The second to top "Related" question (https://economics.stackexchange.com/questions/6908/what-utility-functions-are-equivalent-to-additive-functions?rq=1) contained a link to Ted Bergstrom's [Lecture Notes on Separable Preferences][1], which answered this question in the affirmative. The full proof is given in "Donald W. Katzner. Static Demand Theory. Macmillan, New York, 1970".

  [1]: http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/separabilitynotes.pdf