ZF alone does not prove that every PID is a UFD, according to <a href="http://journals.cambridge.org/action/displayAbstract?aid=2076124">this paper</a>: Hodges, Wilfrid. *Läuchli's algebraic closure of $Q$.* Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 2, 289--297. <a href="http://www.ams.org/mathscinet-getitem?mr=422022">MR 422022</a>.

One result in this paper is the following:<blockquote>COROLLARY 10. Neither (a) nor (b) is provable from ZF alone:<br />
(a) Every principal ideal domain is a unique factorization domain.<br />
(6) Every principal ideal domain has a maximal ideal.</blockquote> 

By the way, I didn't know the answer to this question until today. To find the answer, I consulted Howard and Rubin's book *Consequences of the Axiom of Choice*. (Actually, I did a search for "principal ideal domain" of their book using <a href="http://books.google.co.uk/books?id=YXaVkHPQED4C&lpg=PP1&pg=PP1#v=onepage&q&f=false">Google Books</a>.)