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 $2.3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${D}^{0}\ensuremath{\rightarrow}{K}_{\mathit{S}}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ 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 ${x}_{CP}=[2.7\ifmmode\pm\else\textpm\fi{}1.6(\text{stat})\ifmmode\pm\else\textpm\fi{}0.4(\text{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and the $CP$-violating parameter $\mathrm{\ensuremath{\Delta}}x=[\ensuremath{-}0.53\ifmmode\pm\else\textpm\fi{}0.70(\text{stat})\ifmmode\pm\else\textpm\fi{}0.22(\text{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. 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.