  Using the method of asymptotics beyond all orders, we evaluate the amplitude
of radiation from a moving small-amplitude soliton in the discrete nonlinear
Schr\"odinger equation. When the nonlinearity is of the cubic type, this
amplitude is shown to be nonzero for all velocities and therefore
small-amplitude solitons moving without emitting radiation do not exist. In the
case of a saturable nonlinearity, on the other hand, the radiation is found to
be completely suppressed when the soliton moves at one of certain isolated
`sliding velocities'. We show that a discrete soliton moving at a general speed
will experience radiative deceleration until it either stops and remains pinned
to the lattice, or--in the saturable case--locks, metastably, onto one of the
sliding velocities. When the soliton's amplitude is small, however, this
deceleration is extremely slow; hence, despite losing energy to radiation, the
discrete soliton may spend an exponentially long time travelling with virtually
unchanged amplitude and speed.
