  Bimetric theory describes gravitational interactions in the presence of an
extra spin-2 field. Previous work has suggested that its cosmological solutions
are generically plagued by instabilities. We show that by taking the Planck
mass for the second metric, $M_f$, to be small, these instabilities can be
pushed back to unobservably early times. In this limit, the theory approaches
general relativity with an effective cosmological constant which is,
remarkably, determined by the spin-2 interaction scale. This provides a
late-time expansion history which is extremely close to $\Lambda$CDM, but with
a technically-natural value for the cosmological constant. We find $M_f$ should
be no larger than the electroweak scale in order for cosmological perturbations
to be stable by big-bang nucleosynthesis. We further show that in this limit
the helicity-0 mode is no longer strongly-coupled at low energy scales.
