For number field $K$, is maximal order $O_K$ flat over the order $O$?

Question

I am now trying to solve the following question:

Suppose $L$ and $K$ are number fields, $K\subset L$, $\Gamma$ is the order of $L$, $O\subset O_K$ is the inverse image of $\Gamma$ via the the inclusion of $K\subset L$. Can we conclude that $\Gamma\otimes_O O_K$ is a subring of $O_L\otimes_O O_K$?

This is deduced to consider if $O_K$ is flat over $O$. Note that $O$ is not Dedekind, it seems not easy to show.

Answer 1

The answer is given by Exercise 1.2.10 in Liu's Algebraic Geometry and Arithmetic Curves:

Let $A$ be an integral domain and $B$ its integral closure in the field of fractions $\mathrm{Frac}(A).$ Suppose that $B$ is a finitely generated $A$-module. Show that $B$ is flat over $A$ if and only if $B = A$. One can show this result is true without the assumption of finiteness of $B$ over $A$.

Though I, too, am at a loss as to how to complete this exercise...