Let $\Omega$ be a smoothly bounded convex domain of finite type in $\C^n$. We show that a divisor in $\Omega$ satisfying the Blaschke condition (respectively associated to a current of order $a>0$) can be defined by a function in the Nevanlinna class $N_0 (\Omega)$ (respectively the Nevanlinna-Djrbachian class $N_{a}(\Omega)$). The proof is based on $L^{1}(b\Omega)$ estimates (resp. weighted $L^1 (\Omega) $ estimates) for the solution of the $\bar\partial$-equation on $\Omega$.