Measurements of the thermopower of a one-dimensional (1D) ballistic constriction are presented and fitted by calculations for a constriction where the local potential at the bottleneck forms a saddle. The calculations show that the thermopower can be well approximated by a linear-response formulation far outside its normal range of applicability, provided that an appropriate average temperature is assumed. The fits give a saddle parameter of omega _y/ omega _x>or=2, which sets an upper limit of 2.2 meV on the 1D sub-band spacing.