Confidence intervals for the normal mean utilizing prior information

Consider $X_{1},X_{2},...,X_{n}$ that are independent and identically $\text{N}(\mu,\sigma^{2})$ distributed. Suppose that we have uncertain prior information that $\mu = 0$. We answer the question: to what extent can a frequentist $1-\alpha$ confidence interval for $\mu$ utilize this prior information?