We report on the discovery of strongly variable FeII and MgII absorption lines seen at z=1.48 in the spectra of the z=4.05 GRB 060206 obtained between 4.13 to 7.63 hours (observer frame) after the burst. In particular, the FeII line equivalent width (EW) decayed rapidly from 1.72+-0.25 AA to 0.28+-0.21 AA, only to increase to 0.96+-0.21 AA in a later date spectrum. The MgII doublet shows even more complicated evolution: the weaker line of the doublet drops from 2.05+-0.25 AA to 0.92+-0.32 AA, but then more than doubles to 2.47+-0.41 AA in later data. The ratio of the EWs for the MgII doublet is also variable, being closer to 1:1 (saturated regime) when the lines are stronger and becoming closer to 2:1 (unsaturated regime) when the lines are weaker, consistent with expectations based on atomic physics. We have investigated and rejected the possibility of any instrumental or atmospheric effects causing the observed strong variations. Our discovery of clearly variable intervening FeII and MgII lines lends very strong support to their scenario, in which the characteristic size of intervening patches of MgII ``clouds is comparable to the GRB beam size, i.e, about 10^16 cm. We discuss various implications of this discovery, including the nature of the MgII absorbers, the physics of GRBs, and measurements of chemical abundances from GRB and quasar absorption lines.
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