  The DARWIN collaboration recently argued that DARWIN (DARk matter WImp search
with liquid xenoN) can collect, via neutrino--electron scattering, a large,
useful sample of solar $pp$-neutrinos, and measure their survival probability
with sub-percent precision. We explore the physics potential of such a sample
in more detail. We estimate that, with 300 ton-years of data, DARWIN can also
measure, with the help of current solar neutrino data, the value of
$\sin^2\theta_{13}$, with the potential to exclude $\sin^2\theta_{13}=0$ close
to the three-sigma level. We explore in some detail how well DARWIN can
constrain the existence of a new neutrino mass-eigenstate $\nu_4$ that is
quasi-mass-degenerate with $\nu_1$ and find that DARWIN's sensitivity
supersedes that of all current and near-future searches for new, very light
neutrinos. In particular, DARWIN can test the hypothesis that $\nu_1$ is a
pseudo-Dirac fermion as long as the induced mass-squared difference is larger
than $10^{-13}$ eV$^2$, one order of magnitude more sensitive than existing
constraints. Throughout, we allowed for the hypotheses that DARWIN is filled
with natural xenon or $^{136}$Xe-depleted xenon.
