  Our aim is to construct a functorial tensor product of $A_\infty$-algebras
or, equivalently, an explicit diagonal for the operad of cellular chains, over
the integers, of the Stasheff associahedron. These construction were in fact
already indicated by R. Umble and S. Saneblidze in [9]; we will try to give a
more satisfactory presentation. We also prove that there does not exist an
associative tensor product of $A_\infty$-algebras.
