A study of the decays $B^0\to \mu^+\mu^-$ and $B^0_s\to \mu^+\mu^-$ has been performed using data corresponding to an integrated luminosity of $25$ fb$^{-1}$ of $7$ TeV and $8$ TeV proton--proton collisions collected with the ATLAS detector during the LHC Run 1. For $B^0$, an upper limit on the branching fraction is set at ${\cal B}(B^0 \to \mu^+\mu^-) < 4.2 \times 10^{-10}$ at $95\%$ confidence level. For $B^0_s$, the branching fraction ${\cal B}(B^0_s \to \mu^+\mu^-) = \left( 0.9^{+1.1}_{-0.8} \right) \times 10^{-9}$ is measured. The results are consistent with the Standard Model expectation with a $p$-value of $4.8\%$, corresponding to $2.0$ standard deviations.