  In LEP searches for the neutral CP-odd scalar $A^0$ of a multi-Higgs doublet
model, experimenters have searched for $Z\rightarrow h^0 A^0$ (where $h^0$ is
the lightest CP-even scalar). No model-independent limit on the $A^0$ mass can
be deduced from present data if $m_{h^0}>m_Z$. In this paper, we compute the
rates for $Z\rightarrow A^0 A^0\nu\bar\nu$ and $e^+e^-\rightarrow A^0 A^0 Z$.
Assuming that no light CP-even neutral scalars exist, the branching ratio for
$Z\rightarrow A^0 A^0\nu\bar\nu$ is found to be less than $1.4\times10^{-8}$.
At higher $e^+e^-$ center-of-mass energies, $\sigma(e^+e^-\rightarrow A^0 A^0
Z)$ peaks at $0.5$~fb. A comparison with other direct searches for the $A^0$ is
briefly considered.
