  Using data collected with the BESIII detector in $e^+e^-$ collisions at
center-of-mass energies between 4.178 and 4.226 GeV and corresponding to
6.32~fb$^{-1}$ of integrated luminosity, we report the amplitude analysis and
branching-fraction measurement of the $D^+_s \to \pi^+ \pi^0 \eta^{\prime}$
decay. We find that the dominant intermediate process is $D^+_s \to\rho^+
\eta^{\prime}$ and the significances of other resonant and nonresonant
processes are all less than $3\sigma$. The upper limits on the branching
fractions of $S$-wave and $P$-wave nonresonant components are set to $0.10\%$
and $0.74\%$ at the $90\%$ confidence level, respectively. In addition, the
branching fraction of the $D^+_s \to \pi^+ \pi^0 \eta^{\prime}$ decay is
measured to be $(6.15\pm0.25(\rm stat.)\pm0.18(\rm syst.))\%$, which receives
significant contribution only from $D_s^+\to \rho^+\eta^{\prime}$ according to
the amplitude analysis.

High Energy Physics - Experiment