  We show how to derive exact boundary $S$ matrices for integrable quantum
field theories in 1+1 dimensions using lattice regularization. We do this
calculation explicitly for the sine-Gordon model with fixed boundary conditions
using the Bethe ansatz for an XXZ-type spin chain in a boundary magnetic field.
Our results agree with recent conjectures of Ghoshal and Zamolodchikov, and
indicate that the only solutions to the Bethe equations which contribute to the
scaling limit are the standard strings.
