Let $A$ be a local ring (I am also happy to assume it is Noetherian) with maximal ideal $\mathfrak{m}$.
Claim: If the associated graded ring $$G_{\mathfrak{m}}(A)=\bigoplus_{n\geq 0} \mathfrak{m}^n/\mathfrak{m}^{n+1}$$ is an integrally closed domain, then $A$ is an integrally closed domain.
This claim appears without proof in Chapter 11 of Atiyah-MacDonald (page 123). I see why $A$ must be a domain, but have been unsuccessful in verifying the rest of the claim.
References for a proof or suggestions for how to proceed would be much appreciated.