  Near-future data from ESA's Gaia mission will provide precise, full
phase-space information for hundreds of millions of stars out to heliocentric
distances of ~10 kpc. This "horizon" for full phase-space measurements is
imposed by the Gaia parallax errors degrading to worse than 10%, and could be
significantly extended by an accurate distance indicator. Recent work has
demonstrated how Spitzer observations of RR Lyrae stars can be used to make
distance estimates accurate to 2%, effectively extending the Gaia, precise-data
horizon by a factor of ten in distance and a factor of 1000 in volume. This
Letter presents one approach to exploit data of such accuracy to measure the
Galactic potential using small samples of stars associated with debris from
satellite destruction. The method is tested with synthetic observations of 100
stars from the end point of a simulation of satellite destruction: the shape,
orientation, and depth of the potential used in the simulation are recovered to
within a few percent. The success of this simple test with such a small sample
in a single debris stream suggests that constraints from multiple streams could
be combined to examine the Galaxy's dark matter halo in even more detail --- a
truly unique opportunity that is enabled by the combination of Spitzer and Gaia
with our intimate perspective on the Galaxy.
