  We present the results of our investigation of the multiwavelength data on
HESS J1303-631, an unidentified TeV source serendipitously discovered in the
Galactic plane by the HESS collaboration. These results strongly suggest the
identification of this particular source as the remnant of a Gamma-Ray Burst
(GRB) that happened some few tens of thousands years ago in our Galaxy at a
distance on the order of 10 kpc from us. We show, through detailed calculations
of particle diffusion, interaction and radiation processes of relativistic
particles in the interstellar medium, that it is possible for a GRB remnant
(GRBR) to be a strong TeV emitter with no observable synchrotron emission. We
predict spectral and spatial signatures that would unambiguously distinguish
GRBRs from ordinary supernova remnants, including: (1) large energy budgets
inferred from their TeV emission, but at the same time, (2) suppressed fluxes
in the radio through GeV wavebands; (3) extended center-filled emission with an
energy-dependent spatial profile; and (4) a possible elongation in the
direction of the past pair of GRB jets. While GRBRs can best be detected by
ground-based gamma-ray detectors, the future GLAST mission will play a crucial
role in confirming the predicted low level of GeV emission.
