  We show that if two gradient Ricci solitons are asymptotic along some end of
each to the same regular cone, then the soliton metrics must be isometric on
some neighborhoods of infinity of these ends. Our theorem imposes no
restrictions on the behavior of the metrics off of the ends in question and in
particular does not require their geodesic completeness. As an application, we
prove that the only complete connected gradient shrinking Ricci soliton
asymptotic to a rotationally symmetric cone is the Gaussian soliton on R^n.
