A search for excited states of the $B^{\pm}_c$ meson is performed using 4.9 fb$^{-1}$ of 7 TeV and 19.2 fb$^{-1}$ of 8 TeV $pp$ collision data collected by the ATLAS experiment at the LHC. A new state is observed through its hadronic transition to the ground state, with the latter detected in the decay $B_c^{\pm} \to J/\psi \pi^{\pm}$. The state appears in the $m(B_c^{\pm}\pi^{+}\pi^{-})-m(B_c^{\pm})-2m(\pi^{\pm})$ mass difference distribution with a significance of 5.2 standard deviations. The mass of the observed state is $6842 \pm 4 \pm 5$ MeV, where the first error is statistical and the second is systematic. The mass and decay of this state are consistent with expectations for the second $S\mbox{-}$wave state of the $B^{\pm}_c$ meson, $B^{\pm}_c(2S)$.