  Motivated by recent progress in trapping Bose-Einstein condensed atoms in
toroidal potentials, we examine solitary-wave solutions of the nonlinear
Schr\"odinger equation subject to periodic boundary conditions. When the
circumference of the ring is much larger than the size of the wave, the density
profile is well approximated by that of an infinite ring, however the density
and the velocity of propagation cannot vanish simultaneously. When the size of
the ring becomes comparable to the size of the wave, the density variation
becomes sinusoidal and the velocity of propagation saturates to a constant
value.
