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 2.3×10^{6} D^{0}→K_{S}^{0}π^{+}π^{-} 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±1.6(stat)±0.4(syst)]×10^{-3} and the CP-violating parameter Δx=[-0.53±0.70(stat)±0.22(syst)]×10^{-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.