  Three experimental observations on 1s-excitons in Cu$_2$O are not consistent
with the picture of the exciton as a simple hydrogenic bound state: the
energies of the 1s-excitons deviate from the Rydberg formula, the total exciton
mass exceeds the sum of the electron and hole effective masses, and the
triplet-state excitons lie above the singlet. Incorporating the band structure
of the material, we calculate the corrections to this simple picture arising
from the fact that the exciton Bohr radius is comparable to the lattice
constant. By means of a self-consistent variational calculation of the total
exciton mass as well as the ground-state energy of the singlet and the
triplet-state excitons, we find excellent agreement with experiment.
