Condition for conormal module of commutative, Noetherian, local ring to have finite length

Question

Let $A$ be a commutative, Noetherian, local ring, $O$ a discrete valuation ring and $\lambda : A \rightarrow O$ be an epimorphism. Let $p=\ker(\lambda)$, and consider the conormal $A$-module $p/p^2$. I'm reading that the condition that $p/p^2$ has finite length is equivalent to the natural map $A_p \rightarrow O_{(0)}$ (from the localization of $A$ at $p$ to the field of fractions of $O$) being an isomorphism, but why is this true? Module length is a relatively new topic for me and I cannot understand the equivalence. Thanks in advance!