There is a webpage where you can find mention of this: http://jeff560.tripod.com/set.html
Here's an extract
The null set symbol (Ø) first appeared in N. Bourbaki Éléments de mathématique Fasc.1: Les structures fondamentales de l'analyse; Liv.1: Theorie de ensembles. (Fascicule de resultants) (1939): "certaines propriétés... ne sont vraies pour aucun élément de E... la partie qu’elles définissent est appelée la partie vide de E, et designée par la notation Ø." (p. 4.)
André Weil (1906-1998) says in his autobiography that he was responsible for the symbol:
Wisely, we had decided to publish an installment establishing the system of notation for set theory, rather than wait for the detailed treatment that was to follow: it was high time to fix these notations once and for all, and indeed the ones we proposed, which introduced a number of modifications to the notations previously in use, met with general approval. Much later, my own part in these discussions earned me the respect of my daughter Nicolette, when she learned the symbol Ø for the empty set at school and I told her that I had been personally responsible for its adoption. The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar.
The citation above is from page 114 of André Weil's The Apprenticeship of a Mathematician, Birkhaeuser Verlag, Basel-Boston-Berlin, 1992. Translated from the French by Jennifer Gage. The citation was provided by Julio González Cabillón.
You should also note that Nicolas Bourbaki is a collective pseudonym for a group of French-speaking mathematicians.
Concerning your question, I remember one of my math professor telling us that in fact, it was a claim by both André Weil and Claude Chevalley, but I can't find the citation for Chevalley.
I've found also this lecture about history of math from a Canadian University (in French), where this issue is mentioned: http://www.mat.ulaval.ca/fileadmin/Cours/MAT-2500/Bourbaki.pdf