  We report a measurement of the mass difference between neutral charm-meson
eigenstates using a novel approach that enhances sensitivity to this parameter.
We use $2.3\times 10^6$ $D^0\to K_S^0\pi^+\pi^-$ decays reconstructed in
proton-proton collisions collected by the \lhcb experiment in 2011 and 2012.
Allowing for $CP$ violation in mixing and in the interference between mixing
and decay, we measure the $CP$-averaged normalized mass difference $x_{CP} =
\left[2.7\pm1.6\text{(stat)}\pm0.4\text{(syst)}\right]\times10^{-3}$ and the
$CP$-violating parameter $\Delta x =
\left[-0.53\pm0.70\text{(stat)}\pm0.22\text{(syst)}\right]\times10^{-3}$. The
results are consistent with $CP$ symmetry. These determinations are the most
precise from a single experiment and, combined with current world-average
results, yield the first evidence that the masses of the neutral charm-meson
eigenstates differ.
