  We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a
toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this
classification, we show that any smooth complete toric variety $X$ of dimension
$n\geq 3$ with a $T$-fixed point $x\in X$ such that the blow-up $B_x(X)$ of $X$
at $x$ is Fano is isomorphic either to $\PP^n$ or to the blow-up of $\PP^n$
along a $\PP^{n-2}$. As expected, such results are proved using toric Mori
theory due to Reid.
