  The double-charm tetraquark meson $T_{cc}^+(3875)$ can be produced in
high-energy proton-proton collisions by the creation of the charm mesons
$D^{*+} D^0$ at short distances followed by their binding into $T_{cc}^+$. The
$T_{cc}^+$ can also be produced by the creation of $D^{*+} D^{*+}$ at short
distances followed by their rescattering into $T_{cc}^+ \pi^+$. A charm-meson
triangle singularity produces a narrow peak in the $T_{cc}^+ \pi^+$ invariant
mass distribution 6.1 MeV above the threshold with a width of about 1 MeV. Well
beyond the peak, the differential cross section decreases with the invariant
kinetic energy $E$ of $T_{cc}^+ \pi^+$ as $E^{-1/2}$. The fraction of
$T_{cc}^+$ that are accompanied by $\pi^+$ with $E< m_\pi$ is estimated to be
roughly 3%. The fraction of $T_{cc}^+$ events with $T_{cc}^+ \pi^+$ in the
narrow peak from the triangle singularity could be comparable.
