  Let $(\az,F)$ be a bipermutative algebraic cellular automaton. We present
conditions which force a probability measure which is invariant for the
$\N\times\Z$-action of $F$ and the shift map $\s$ to be the Haar measure on
$\gs$, a closed shift-invariant subgroup of the Abelian compact group $\az$.
This generalizes simultaneously results of B. Host, A. Maass and S.
Mart\'{\i}nez \cite{Host-Maass-Martinez-2003} and M. Pivato \cite{Pivato-2003}.
This result is applied to give conditions which also force a $(F,\s)$-invariant
probability measure to be the uniform Bernoulli measure when $F$ is a
particular invertible expansive cellular automaton on $\an$.
