  We investigate the detectability of a proposed population of Gamma-Ray Bursts
(GRBs) from the collapse of Population III (Pop III) stars. The James Webb
Space Telescope (JWST) and Space Infrared telescope for Cosmology and
Astrophysics (SPICA) will be able to observe the late time infrared afterglows.
We have developed a new method to calculate their detectability, which takes
into account the fundamental initial mass function (IMF) and formation rates of
Pop III stars, from which we find the temporal variability of the afterglows
and ultimately the length of time JWST and SPICA can detect them. In the range
of plausible Pop III GRB parameters, the afterglows are always detectable by
these instruments during the isotropic emission, for a minimum of 55 days and a
maximum of 3.7 years. The average number of detectable afterglows will be
2.96$\times 10^{-5}$ per SPICA field of view (FOV) and 2.78$\times 10^{-6}$ per
JWST FOV. These are lower limits, using a pessimistic estimate of Pop III star
formation. An optimal observing strategy with SPICA could identify a candidate
orphan afterglow in $\sim 1.3$ years, with a 90 percent probability of
confirmation with further detailed observations. A beamed GRB will align with
the FOV of the planned GRB detector Energetic X-ray Imaging Survey Telescope
once every 9 years. Pop III GRBs will be more easily detected by their
isotropic emissions (i.e. orphan afterglows) rather than by their prompt
emissions.
