  We extend the second part of \cite{Bre20} on the uniqueness of ancient
$\kappa$-solutions to higher dimensions. In dimensions $n \geq 4$, an ancient
$\kappa$-solution is a nonflat, complete, ancient solution of the Ricci flow
that is uniformly PIC and weakly PIC2; has bounded curvature; and is
$\kappa$-noncollapsed. We show that the only noncompact ancient
$\kappa$-solutions up to isometry are a family of shrinking cylinders, a
quotient thereof, or the Bryant soliton.
