In fact the following is true.

Proposition: Let $\mathcal{O}_K$ be the ring of integers of a number field $K$. For any $p_0$, it is possible to find finitely many primes $p_1, ..., p_n \ge p_0$ such that $\mathcal{O}_K[p_1^{-1}, ..., p_n^{-1}]$ is a PID.

Proof. Find ideals $I_1, ..., I_m$ representing every ideal class in the ideal class group of $\mathcal{O}_K$ whose norms are not divisible by primes less than $p_0$ and invert all primes dividing their norms. $\Box$

In fact the following is true.

Proposition: Let $\mathcal{O}_K$ be the ring of integers of a number field $K$. For any $p_0$, it is possible to find finitely many primes $p_1, ..., p_n \ge p_0$ such that $\mathcal{O}_K[p_1^{-1}, ..., p_n^{-1}]$ is a PID.

Proof. Find ideals $I_1, ..., I_m$ representing every ideal class in the ideal class group of $\mathcal{O}_K$ whose norms are not divisible by primes less than $p_0$ and invert all primes dividing their norms. $\Box$

Edit (12/4/13): Regarding the last line, every Dedekind domain is an inverse limit of UFDs. Indeed, every Dedekind domain is the inverse limit of its localizations at every maximal ideal, and these are all DVRs (in particular PIDs, in particular UFDs).