  We present ~ 126 new spectroscopically identified members of the GD-1 tidal
stream obtained with the 5000-fiber Dark Energy Spectroscopic Instrument
(DESI). We confirm the existence of a ``cocoon'' which is broad
(FWHM~2.932deg~460pc) and kinematically hot (velocity dispersion,
sigma~5-8km/s) component that surrounds a narrower (FWHM~0.353deg~55pc) and
colder (sigma~ 2.2-2.6km/s) thin stream component (based on a median per star
velocity precision of 2.7km/s). The cocoon extends over at least a ~ 20deg
segment of the stream observed by DESI. The thin and cocoon components have
similar mean values of [Fe/H]: -2.54+/- 0.04dex and -2.45+/-0.06dex suggestive
of a common origin. The data are consistent with the following scenarios for
the origin of the cocoon. The progenitor of the GD-1 stream was an accreted
globular cluster (GC) and: (a) the cocoon was produced by pre-accretion tidal
stripping of the GC while it was still inside its parent dwarf galaxy; (b) the
cocoon is debris from the parent dwarf galaxy; (c) an initially thin GC tidal
stream was heated by impacts from dark subhalos in the Milky Way; (d) an
initially thin GC stream was heated by a massive Sagittarius dwarf galaxy; or a
combination of some these. In the first two cases the velocity dispersion and
mean metallicity are consistent with the parent dwarf galaxy having a halo mass
of ~0^9\msun. Future DESI spectroscopy and detailed modeling may enable us to
distinguish between these possible origins.
