  We show that an eternal solution to a complete, locally conformally flat
Yamabe flow, $\frac{\partial}{\partial t} g = -Rg$, with uniformly bounded
scalar curvature and positive Ricci curvature at $t = 0$, where the scalar
curvature assumes its maximum is a gradient steady soliton. As an application
of that, we study the blow up behavior of $g(t)$ at the maximal time of
existence, $T < \infty$. We assume that $(M,g(\cdot, t))$ satisfies (i) the
injectivity radius bound {\bf or} (ii) the Schouten tensor is positive at time
$t = 0$ and the scalar curvature bounded at each time-slice. We show that the
singularity the flow develops at time $T$ is always of type I.
