  We describe a search for X-ray afterglows from gamma-ray bursts using the
ROSAT all-sky survey (RASS) data. If the emission in the soft X-ray band is
significantly less beamed than in the gamma-ray band, we expect to detect many
afterglows in the RASS. We found 23 afterglow candidates, where about 4
detections are predicted. However, follow-up spectroscopy of several
counterpart candidates strongly suggests a flare star origin of the RASS events
in many cases. We conclude that the RASS data are consistent with comparable
beaming angles in the X-ray (within a few hours of the burst) and gamma-ray
bands. However, models predicting a large amount of energy emerging as a nearly
isotropic X-ray component of the early afterglow are severely constrained by
the ROSAT data. In particular, a so far undetected class of ``dirty fireballs''
and delayed ``rebursts'' are constrained.
