  Discovering neutrino decay would be strong evidence of physics beyond the
Standard Model. Presently, there are only lax lower limits on the lifetime
$\tau$ of neutrinos, of $\tau/m \sim 10^{-3}$ s eV$^{-1}$ or worse, where $m$
is the unknown neutrino mass. High-energy cosmic neutrinos, with TeV-PeV
energies, offer superior sensitivity to decay due to their cosmological-scale
baselines. To tap into it, we employ a promising method, recently proposed,
that uses the Glashow resonance $\bar{\nu}_e + e \to W$, triggered by
$\bar{\nu}_e$ of 6.3 PeV, to test decay with only a handful of detected events.
If most of the $\nu_1$ and $\nu_2$ decay into $\nu_3$ en route to Earth, no
Glashow resonance would occur in neutrino telescopes, because the remaining
$\nu_3$ have only a tiny electron-flavor content. We turn this around and use
the recent first detection of a Glashow resonance candidate in IceCube to place
new lower limits on the lifetimes of $\nu_1$ and $\nu_2$. For $\nu_2$, our
limit is the current best. For $\nu_1$, our limit is close to the current best
and, with the imminent detection of a second Glashow resonance, will vastly
surpass it.
