  Let $S$ and $T$ be reduced divisors on $\mathbb{P}^2$ which have no common
components, and $\Delta=S+2\,T.$ We assume $\deg\Delta=6.$ Let
$\pi:X\to\mathbb{P}^2$ be a normal triple cover with branch divisor $\Delta,$
i.e. $\pi$ is ramified along $S$ (resp. $T$) with the index 2 (resp. 3). In
this note, we show that $X$ is either a $\mathbb{P}^1$-bundle over an elliptic
curve or a normal cubic surface in $\mathbb{P}^3.$ Consequently, we give a
necessary and sufficient condition for $\Delta$ to be the branch divisor of a
normal triple cover over $\mathbb{P}^2.$
