  Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on
$\mathbb P^2$ and $V_{\alpha}(T)$ be the upper level sets of Lelong numbers
$\nu(T,x)$ of $T$. For any $\alpha\geq \frac{1}{3}$, we show that
$|V_{\alpha}(T)\setminus C|\leq 2$ for some cubic curve $C$.
