  The observation of neutrinoless double-beta ($0\nu\beta\beta$) decay would
offer proof of lepton number violation, demonstrating that neutrinos are
Majorana particles, while also helping us understand why there is more matter
than antimatter in the Universe. If the decay is driven by the exchange of the
three known light neutrinos, a discovery would, in addition, link the observed
decay rate to the neutrino mass scale through a theoretical quantity known as
the nuclear matrix element (NME). Accurate values of the NMEs for all nuclei
considered for use in $0\nu\beta\beta$ experiments are therefore crucial for
designing and interpreting those experiments. Here, we report the first
comprehensive ab initio uncertainty quantification of the
$0\nu\beta\beta$-decay NME, in the key nucleus $^{76}$Ge. Our method employs
nuclear strong and weak interactions derived within chiral effective field
theory and recently developed many-body emulators. Our result, with a
conservative treatment of uncertainty, is an NME of $2.60^{+1.28}_{-1.36}$,
which, together with the best-existing half-life sensitivity and phase-space
factor, sets an upper limit for effective neutrino mass of $187^{+205}_{-62}$
meV. The result is important for designing next-generation germanium detectors
aiming to cover the entire inverted hierarchy region of neutrino masses.
