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. $\Box$