$$\left( \sum_{g \in G}m_gg\right)\cdot a = \sum_{g \in G}m_g\rho(g)(a)$$
But this action isn't clear to me, for example it isn't stated what is meant by $m_g$ and it seems like we have a possibly infinite (depending on the order of $G$) sum in the brackets whereas $\mathbb{Z}G$ has only finite sums.