Going-down theorem

Let $R\subseteq A$ be an integral extension of domains with $R$ integrally closed. Given primes ${\mathfrak{p}}_0\subseteq {\mathfrak{p}}_1$ of $R$ and a prime ${\mathfrak{q}}_1$ of $A$ over ${\mathfrak{p}}_1$, there is a prime ${\mathfrak{q}}_0\subseteq {\mathfrak{q}}_1$ over ${\mathfrak{p}}_0$.