We combine Spitzer and ground-based observations to measure the microlens parallax of OGLE-2005-SMC-001, the first such space-based determination since S. Refsdal proposed the idea in 1966. The parallax measurement yields a projected velocity ṽ~230 km s-1, the typical value expected for halo lenses, but an order of magnitude smaller than would be expected for lenses lying in the Small Magellanic Cloud itself. The lens is a weak (i.e., non-caustic-crossing) binary, which complicates the analysis considerably but ultimately contributes additional constraints. Using a test proposed by Assef and coworkers, which makes use only of kinematic information about different populations but does not make any assumptions about their respective mass functions, we find that the likelihood ratio is Lhalo/LSMC=20. Hence, halo lenses are strongly favored, but Small Magellanic Cloud (SMC) lenses are not definitively ruled out. Similar Spitzer observations of additional lenses toward the Magellanic Clouds would clarify the nature of the lens population. The Space Interferometry Mission could make even more constraining measurements.