  In this paper we classify Euclidean hypersurfaces $f\colon M^n \rightarrow
\mathbb{R}^{n+1}$ with a principal curvature of multiplicity $n-2$ that admit a
genuine conformal deformation $\tilde{f}\colon M^n \rightarrow
\mathbb{R}^{n+2}$. That $\tilde{f}\colon M^n \rightarrow \mathbb{R}^{n+2}$ is a
genuine conformal deformation of $f$ means that it is a conformal immersion for
which there exists no open subset $U \subset M^n$ such that the restriction
$\tilde{f}|_U$ is a composition $\tilde f|_U=h\circ f|_U$ of $f|_U$ with a
conformal immersion $h\colon V\to \mathbb{R}^{n+2}$ of an open subset $V\subset
\mathbb{R}^{n+1}$ containing $f(U)$.
