  Topological magnons are bosonic analogues of topological fermions in
electronic systems. They have been studied extensively by theory but rarely
realized by experiment. Here, by performing inelastic neutron scattering
measurements on single crystals of a two-dimensional ferromagnet CrBr$_3$,
which was classified as Dirac magnon semimetal featured by the linear bands
crossing at the Dirac points, we fully map out the magnetic excitation spectra,
and reveal that there is an apparent gap of $\sim$3.5~meV between the acoustic
and optical branches of the magnons at the K point. By collaborative efforts
between experiment and theoretical calculations using a five-orbital Hubbard
model obtained from first-principles calculations to derive the exchange
parameters, we find that a Hamiltonian with Heisenberg exchange interactions,
next-nearest-neighbor Dzyaloshinskii-Moriya (DM) interaction, and single-ion
anisotropy is more appropriate to describe the system. Calculations using the
model show that the lower and upper magnon bands separated by the gap exhibit
Chern numbers of $\pm1$. These results indicate that CrBr$_3$ is a topological
magnon insulator, where the nontrivial gap is a result of the DM interaction.
