  Neutrinos mix and have mass differences, so decays from one to another must
occur. But how fast? The best direct limits on non-radiative decays, based on
solar and atmospheric neutrinos, are weak, $\tau \gtrsim 10^{-3}$ s ($m$/eV) or
much worse. Greatly improved sensitivity, $\tau \sim 10^3$ s ($m$/eV), will
eventually be obtained using neutrinos from distant astrophysical sources, but
large uncertainties --- in neutrino properties, source properties, and
detection aspects --- do not allow this yet. However, there is a way forward
now. We show that IceCube diffuse neutrino measurements, supplemented by
improvements expected in the near term, can increase sensitivity to $\tau \sim
10$ s ($m$/eV) for all neutrino mass eigenstates. We provide a roadmap for the
necessary analyses and show how to manage the many uncertainties. If limits are
set, this would definitively rule out the long-considered possibility that
neutrino decay affects solar, atmospheric, or terrestrial neutrino experiments.

High Energy Astrophysical Phenomena