The first observation of a new meson state decaying into $B_c^{+}$ and a photon is presented using a dataset of $pp$ collisions at a centre-of-mass energy $\sqrt{s}=13$ TeV collected by the ATLAS detector at the Large Hadron Collider, corresponding to an integrated luminosity of 140 fb${}^{-1}$. The $B_c^{+}$ mesons are reconstructed from a three-muon final state from $B_c^{+}\to J/\psi ({\mu }^{+}{\mu }^{-}){\mu }^{+}{\nu }_{\mu }X$ decays, along with photons reconstructed via conversions to electron–positron pairs in the detector material. The new state is observed with a significance that exceeds 8 standard deviations. The mass difference between the new meson state and the ground-state $B_c^{+}$ meson is measured to be $64.5\pm 1.4{\mathrm{\text{(stat.)}}}_{-1.4}^{+1.0}\mathrm{\text{(syst.)}}$ MeV. This corresponds to a mass for the observed state of $6339.0\pm 1.4{\mathrm{\text{(stat.)}}}_{-1.4}^{+1.0}\mathrm{\text{(syst.)}}\pm 0.3(m_{B_c^{+}})$ MeV, where the last uncertainty is due to the precision of the best knowledge of the $B_c^{+}$ meson mass. The low mass difference value matches the theory expectations for the lowest vector state of the $\bar{b}c$ quarkonium system, the $B_c^{*+}$ meson.