  For a complete, stable theory $T$ we construct, in a reasonably canonical
way, a related stable theory $T^*$ which has higher independent amalgamation
properties over the algebraic closure of the empty-set. The theory $T^*$ is an
algebraic cover of $T$ and we give an explicit description of the finite covers
involved in the construction of $T^*$ from $T$. This follows an approach of E.
Hrushovski. If $T$ is almost strongly minimal with a $0$-definable strongly
minimal set, then we show that $T^*$ has higher amalgamation over any
algebraically closed subset.
