  The optical properties of atomically thin transition metal dichalcogenide
(TMDC) semiconductors are shaped by the emergence of correlated many-body
complexes due to strong Coulomb interaction. Exceptional electron-hole exchange
predestines TMDCs to study fundamental and applied properties of Coulomb
complexes such as valley depolarization of excitons and fine-structure
splitting of trions. Biexcitons in these materials are less understood and it
has been established only recently that they are spectrally located between
exciton and trion.
  Here we show that biexcitons in monolayer TMDCs exhibit a distinct fine
structure on the order of meV due to electron-hole exchange. Ultrafast
pump-probe experiments on monolayer WSe$_2$ reveal decisive biexciton
signatures and a fine structure in excellent agreement with a microscopic
theory. We provide a pathway to access biexciton spectra with unprecedented
accuracy, which is valuable beyond the class of TMDCs, and to understand even
higher Coulomb complexes under the influence of electron-hole exchange.
