In this note, we prove that on an $n$-dimensional compact toric manifold with
positive first Chern class, the K\"ahler-Ricci flow with any initial
$(S^1)^n$-invariant K\"ahler metric converges to a K\"ahler-Ricci soliton. In
particular, we give another proof for the existence of K\"ahler-Ricci solitons
on a compact toric manifold with positive first Chern class by using the
K\"ahler-Ricci flow.