  The critical radius of a core-shell-type nucleus grown by diffusion in a
phase-separated solution is studied. A {\it kinetic} critical radius rather
than the {\it thermodynamic} critical radius of standard classical nucleation
theory can be defined from the diffusional growth equations. It is shown that
there exist two kinetic critical radii for the core-shell-type nucleus, for
which both the inner core radius and the outer shell radius will be stationary.
Therefore, these two critical radii correspond to a single critical point of
the nucleation path with a single energy barrier even though the nucleation
looks like a two-step process. The two radii are given by formulas similar to
that of classical nucleation theory if the Ostwald-Freundlich boundary
condition is imposed at the surface of the inner nucleus and that of the outer
shell. The subsequent growth of a core-shell-type post-critical nucleus follows
the classical picture of Ostwald's step rule. Our result is consistent with
some of the experimental and numerical results which suggest the
core-shell-type critical nucleus.
