  We explore the cosmological signals of theories in which the neutrinos decay
into invisible dark radiation after becoming non-relativistic. We show that in
this scenario, near-future large scale structure measurements from the Euclid
satellite, when combined with cosmic microwave background data from Planck, may
allow an independent determination of both the lifetime of the neutrinos and
the sum of their masses. These parameters can be independently determined
because the Euclid data will cover a range of redshifts, allowing the growth of
structure over time to be tracked. If neutrinos are stable on cosmological
timescales, these observations can improve the lower limit on the neutrino
lifetime by seven orders of magnitude, from $\mathcal{O}(10)$ years to $2\times
10^8$ years ($95\%$ C.L.), without significantly affecting the measurement of
neutrino mass. On the other hand, if neutrinos decay after becoming
non-relativistic but on timescales less than $\mathcal{O}(100)$ million years,
these observations may allow, not just the first measurement of the sum of
neutrino masses, but also the determination of the neutrino lifetime from
cosmology.
