In Section 2 of 'Local Fields : Jean-Pierre Serre', the author proves the following proposition (Propositions are numbered following the book) :
$\textbf{Proposition 3:}$ Let $A$ be a Noetherian integral domain. In order that $A$ be a discrete valuation ring, it is neccessary and sufficient that it satisfy the two following conditions:
(i)$A$ is integrally closed.
(ii) $A$ has a unique non-zero prime ideal.
The author takes about a page to prove the sufficient condition of the proposition, while it seems to me that it is immediate from the proposition just above this proposition, which I mention below :
$\textbf{Proposition:}$ Let $A$ be a commutative ring. In order that $A$ be a discrete valuation ring, it is neccessary and sufficient that it be a Noetherian local ring, and that its maximal ideal be generated by a non-nilpotent element.
Now, if $A$ satisfies (ii) of $\text{Proposition 3:}$, then it is commutative (being an integral domain), Noetherian by assumption, local (since unique non-zero prime ideal $\implies$ unique maximal ideal) and the maximal ideal is obviously generated by a non-nilpotent element, since only nilpotent element in an integral domain is $0$.