  The $\eta^\prime g^* g^{(*)}$ effective vertex function is calculated in the
QCD hard-scattering approach, taking into account the $\eta^\prime$-meson mass.
We work in the approximation in which only one non-leading Gegenbauer moment
for both the quark-antiquark and the gluonic light-cone distribution amplitudes
for the $\eta^\prime$-meson is kept. The vertex function with one off-shell
gluon is shown to have the form (valid for $| q_1^2 | > m_{\eta^\prime}^2$)
$F_{\eta^\prime g^* g} (q_1^2, 0, m_{\eta^\prime}^2) = m_{\eta^\prime}^2
H(q_1^2)/(q_1^2 - m_{\eta^\prime}^2)$, where $H(q_1^2)$ is a slowly varying
function, derived analytically in this paper. The resulting vertex function is
in agreement with the phenomenologically inferred form of this vertex obtained
from an analysis of the CLEO data on the $\eta^\prime$-meson energy spectrum in
the decay $\Upsilon(1S) \to \eta^\prime X$. We also present an interpolating
formula for the vertex function $F_{\eta^\prime g^* g} (q_1^2, 0,
m_{\eta^\prime}^2)$ for the space-like region of the virtuality $q_1^2$, which
satisfies the QCD anomaly normalization for on-shell gluons and the
perturbative-QCD result for the gluon virtuality $| q_1^2| \gtrsim 2$ GeV$^2$.
