  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 $20-60\%$.
