  Neutrinoless double beta decay ($0\nu\beta\beta$) is a yet unobserved nuclear
process which would demonstrate Lepton Number violation, a clear evidence of
beyond Standard Model physics. The process two neutrino double beta decay
($2\nu\beta\beta)$ is allowed by the Standard Model and has been measured in
numerous experiments. In this letter, we report a measurement of
$2\nu\beta\beta$ decay half-life of $^{100}$Mo to the ground state of
$^{100}$Ru of
$(7.07~\pm~0.02~\text{(stat.)}~\pm~0.11~\text{(syst.)})~\times~10^{18}$~yr by
the CUPID-Mo experiment. With a relative precision of $\pm~1.6$ \% this is the
most precise measurement to date of a $2\nu\beta\beta$ decay rate in
$^{100}$Mo. In addition, we constrain higher-order corrections to the spectral
shape which provides complementary nuclear structure information. We report a
novel measurement of the shape factor $\xi_{3,1}=0.45~\pm 0.03~\text{(stat.)} \
\pm 0.05 \ \text{(syst.)}$, which is compared to theoretical predictions for
different nuclear models. We also extract the first value for the effective
axial vector coupling constant obtained from a spectral shape study of
$2\nu\beta\beta$ decay.
