Let $��$ be a self-affine measure on $\mathbb{R}^{d}$ associated to a self-affine IFS $\{��_��(x) = A_��x + v_��\}_{��\in��}$ and a probability vector $p=(p_��)_��>0$. Assume the strong separation condition holds. Let $��_{1}\ge...\ge��_{d}$ and $D$ be the Lyapunov exponents and dimension corresponding to $\{A_��\}_{��\in��}$ and $p^{\mathbb{N}}$, and let $\mathbf{G}$ be the group generated by $\{A_��\}_{��\in��}$. We show that if $��_{m+1}>��_{m}=...=��_{d}$, if $\mathbf{G}$ acts irreducibly on the vector space of alternating $m$-forms, and if the Furstenberg measure $��_{F}$ satisfies $\dim_{H}��_{F}+D>(m+1)(d-m)$, then $��$ is exact dimensional with $\dim��=D$.