  Approaching a quantum critical point (QCP) has been an effective route to
stabilize superconductivity. While the role of magnetic QCPs has been
extensively discussed, similar exploration of a structural QCP is scarce due to
the lack of suitable systems with a continuous structural transition that can
be conveniently tuned to 0~K. Using inelastic X-ray scattering, we examine the
phonon spectrum of the nonmagnetic quasi-skutterudite
(Ca$_{x}$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$, which represents a precious system to
explore the interplay between structural instabilities and superconductivity by
tuning the Ca concentration $x$. We unambiguously detect the softening of
phonon modes around the M point on cooling towards the structural transition.
Intriguingly, at $x=0.85$, the soft mode energy squared at the M point
extrapolates to zero at $(-5.7 \pm 7.7)$~K, providing the first compelling
microscopic evidence of a structural QCP in
(Ca$_{x}$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$. The enhanced phonon density-of-states
at low energy provides the essential ingredient for realizing strong-coupling
superconductivity near the structural QCP.
