Sufficient conditions for quotient ring to be Cohen-Macaulay

Question

We know that every Noetherian integral domain with (Krull) dimension $1$ is Cohen-Macaulay (CM). In a commutative algebra text the author have presented the following problem: "Let $(R,m)$ be a CM local Noetherian ring with a prime ideal $P$ having dimension $1$. Prove that $(R/P,m/P)$ is CM." I say $R/P$ is a Noetherian integral domain with dimension $1$, since $dim(P)=1$. Hence it is CM. Am I right, or the condition that $(R,m)$ being CM is necessary to be given? I appreciate in advance for any help or suggestion.