  We consider the Ricci flow on $\mathbb{CP}^n$ blown-up at one point starting
with any $U(n)$-invariant K\"ahler metric. It is known that the K\"ahler-Ricci
flow must develop Type I singularities. We show that if the total volume does
not go to zero at the singular time, then any Type I parabolic blow-up limit of
the Ricci flow along the exceptional divisor is the unique $U(n)$-complete
shrinking K\"ahler-Ricci soliton on $\mathbb C^n$ blown-up at one point. This
establishes the conjecture of Feldman-Ilmanen-Knopf.
