  Let $G$ be a graph and $C$ a finite set of colours. A vertex colouring
$f:V(G)\to C$ is complete provided that for any two distinct colours
$c_1,c_2\in C$ there is $v_1v_2\in E(G)$ such that $f(v_i)=c_i$, $i=1,2$. The
achromatic number of $G$ is the maximum number $\mathrm{achr}(G)$ of colours in
a proper complete vertex colouring of $G$. In the paper it is proved that if
$q\ge41$ is an odd integer, then the achromatic number of the Cartesian product
of $K_6$ and $K_q$ is $2q+3$.
