Prime ideal in Dedekind domain

Question

Let $\mathcal{o}$ be a Dedekind domain, $K$ its field of fractions, $L$ a finite separable field extension of $K$, and $\mathcal{O}$ the integral closure of $\mathcal{o}$ in $L$. When $\mathfrak{p}$ is a prime ideal in the Dedekind domain $\mathcal{o}$, then one can consider the ideal $\mathfrak{p}\mathcal{O}$ in the Dedekind domain $\mathcal{O}$, and this is sometimes abbreviated as simply $\mathfrak{p}$. What is the relation between $\mathfrak{p}$ and $\mathfrak{p}\mathcal{O}$? Are they in 1-1 correspondence? The latter is usually not a prime ideal in $\mathcal{O}$, right?