A search for CP violation in $D^0 \rightarrow K^{-} K^{+} $ and $D^0 \rightarrow \pi^{-} \pi^{+} $ decays is performed using $pp$ collision data, corresponding to an integrated luminosity of $3~fb^{-1}$, collected using the LHCb detector at centre-of-mass energies of 7 and $8~$TeV. The flavour of the charm meson is inferred from the charge of the pion in $D^{*+}\rightarrow D^0\pi^+$ and $D^{*-}\rightarrow \bar{D^0}\pi^{-}$ decays. The difference between the CP asymmetries in $D^0 \rightarrow K^{-} K^{+} $ and $D^0 \rightarrow \pi^{-} \pi^{+} $ decays, $\Delta A_{CP} \equiv A_{CP}(K^{-} K^{+}) - A_{CP}(\pi^{-} \pi^{+})$, is measured to be $\left( -0.10 \pm 0.08(stat) \pm 0.03(syst) \right) \%$. This is the most precise measurement of a time-integrated CP asymmetry in the charm sector from a single experiment.