There are observational facts and theoretical arguments for an origin of gamma-ray bursts in molecular clouds in distant galaxies. If this is true, one could detect a significant flux of GRB prompt and early afterglow X-ray radiation scattered into our line of sight by the molecular and atomic matter located within tens of parsecs of the GRB site long after the afterglow has faded away. The scattered flux directly measures the typical density of the GRB ambient medium. Furthemore, if the

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... emission is beamed, the scattered X-ray flux will be slowly decreasing for several months to years before falling off rapidly. Therefore, it should be possible to estimate the collimation angle of a burst from the light curve of its X-ray echo and a measured value of the line-of-sight absorption column depth. It is shown that detection of such an echo is for the brightest GRBs just within the reach of the Chandra and XMM-Newton observatories.