  Signals consistent with the B$^+_\mathrm{c}$(2S) and B$^{*+}_\mathrm{c}$(2S)
states are observed in proton-proton collisions at $\sqrt{s} =$ 13 TeV, in an
event sample corresponding to an integrated luminosity of 143 fb$^{-1}$,
collected by the CMS experiment during the 2015-2018 LHC running periods. These
excited $\bar{\mathrm{b}}$c states are observed in the
B$^+_\mathrm{c}\pi^+\pi^-$ invariant mass spectrum, with the ground state
B$^+_\mathrm{c}$ reconstructed through its decay to J/$\psi\,\pi^+$. The two
states are reconstructed as two well-resolved peaks, separated in mass by 29.1
$\pm$ 1.5 (stat) $\pm$ 0.7 (syst) MeV. The observation of two peaks, rather
than one, is established with a significance exceeding five standard
deviations. The mass of the B$^+_\mathrm{c}$(2S) meson is measured to be 6871.0
$\pm$ 1.2 (stat) $\pm$ 0.8 (syst) $\pm$ 0.8 (B$^+_\mathrm{c}$) MeV, where the
last term corresponds to the uncertainty in the world-average B$^+_\mathrm{c}$
mass.
