  The transverse-target single-spin asymmetry for inclusive deep-inelastic
scattering with effectively unpolarized electron and positron beams off a
transversely polarized hydrogen target was measured, with the goal of searching
for a two-photon exchange signal in the kinematic range 0.007 < x_B < 0.9 and
0.25 GeV**2 < Q**2 < 20 GeV**2. In two separate regions Q**2 > 1 GeV**2 and
Q**2 < 1 GeV**2, and for both electron and positron beams, the asymmetries are
found to be consistent with zero within statistical and systematic
uncertainties, which are of order 10**(-3) for the asymmetries integrated over
x_B.
