  Using an {\it ab initio} approach, we report a phonon soft mode in the
tetragonal structure described by the space group $I4_{1}22$ of the $1$ K $5d$
superconductor Cd$_2$Re$_2$O$_7$. It induces an orthorhombic distortion to a
crystal structure described by the space group $F222$ which hosts the
superconducting state. This new phase has a lower total energy than the other
known crystal structures of Cd$_2$Re$_2$O$_7$. Comprehensive temperature
dependent Raman scattering experiments on isotope enriched samples,
$^{116}$Cd$_2$Re$_2{^{18}}$O$_7$, not only confirm the already known structural
phase transitions but also allow us to identify a new characteristic
temperature regime around $\sim 80$ K, below which the Raman spectra undergo
remarkable changes with the development of several sharp modes and mode
splitting. Together with the results of the \textit{ab initio} phonon
calculations we take these observations as strong evidence for another phase
transition to a novel low-temperature crystal structure of Cd$_2$Re$_2$O$_7$.
