The requirement that a (non-Einstein) K\"ahler metric in any given
complex dimension $\,m>2\,$ be almost-everywhere conformally Einstein turns
out to be much more restrictive, even locally, than in the case of complex
surfaces. The local biholomorphic-isometry types of such metrics depend, for
each $\,m>2$, on three real parameters along with an arbitrary
K\"ahler-Einstein metric $\,h\,$ in complex dimension $\,m-1$. We provide an
explicit description of all these local-isometry types, for any
given $\,h$. That result is derived from a more general local classification
theorem for metrics admitting functions we call {\it special K\"ahler-Ricci
potentials}