  The first observation of the $B_c^+ \to J/\psi \pi^+ \pi^0$ decay is reported
with high significance using proton-proton collision data, corresponding to an
integrated luminosity of 9fb$^{-1}$, collected with the LHCb detector at
centre-of-mass energies of 7, 8, and 13 TeV. The ratio of its branching
fraction relative to the $B_c^+ \to J/\psi \pi^+$ channel is measured to be
  $$
  \frac{ {\cal{B}}( B_c^+ \to J/\psi \pi^+\pi^0 ) }
  { {\cal{B}}( B_c^+ \to J/\psi \pi^+ ) }
  = 2.80 \pm 0.15 \pm 0.11 \pm 0.16 \,,
  $$ where the first uncertainty is statistical, the second systematic and the
third related to imprecise knowledge of the branching fractions for $B^+ \to
J/\psi K^{*+}$ and $B^+ \to J/\psi K^+$ decays, which are used to determine the
$\pi^0$ detection efficiency. The $\pi^+\pi^0$ mass spectrum is found to be
consistent with the dominance of an intermediate $\rho^+$ contribution in
accordance with a model based on QCD factorisation.
