  It is proved that, for a prime $p>2$ and integer $n\geq 1$, finite $p$-groups
of nilpotency class $3$ and having only two conjugacy class sizes $1$ and $p^n$
exist if and only if $n$ is even; moreover, for a given even positive integer,
such a group is unique up to isoclinism (in the sense of Philip Hall).
