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 × 106 D0 → K0 Sπþπ− 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 xCP ¼ ½2.7 1.6ðstatÞ 0.4ðsystÞ × 10−3 and the CP-violating parameter Δx ¼ ½−0.53 0.70ðstatÞ 0.22ðsystÞ × 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.