The quark-gluon matter produced in relativistic heavy-ion collisions may contain local domains in which P and CP symmetries are not preserved. When coupled with an external magnetic field, such P- and CP-odd domains will generate electric currents along the magnetic field --- a phenomenon called the chiral magnetic effect (CME). Recently, the STAR Collaboration at RHIC and the ALICE Collaboration at the LHC released data of charge-dependent azimuthal-angle correlators with features consistent with the CME expectation. However, the experimental observable is contaminated with significant background contributions from elliptic-flow-driven effects, which makes the interpretation of the data ambiguous. In this Letter, we show that the collisions of isobaric nuclei, $^{96}_{44}$Ru + $^{96}_{44}$Ru and $^{96}_{40}$Zr + $^{96}_{40}$Zr, provide an ideal tool to disentangle the CME signal from the background effects. Our simulation demonstrates that the two collision types at $sqrt{s_{rm NN}}=200$ GeV have more than $10%$ difference in the CME signal and less than $2%$ difference in the elliptic-flow-driven backgrounds for the centrality range of $20-60%$.
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