While a white dwarf (WD) is, from a theoretical perspective, the most plausible primary star of a Type Ia supernova (SN Ia), many other candidates have not been formally ruled out. Shock energy deposited in the envelope of any exploding primary contributes to the early SN brightness and, since this radiation energy is degraded by expansion after the explosion, the diffusive luminosity depends on the initial primary radius. We present a new non-detection limit of the nearby SN Ia 2011fe, obtained at a time that appears to be just 4 hr after explosion, allowing us to directly constrain the initial primary radius (R$_p$ ). Coupled with the non-detection of a quiescent X-ray counterpart and the inferred synthesized $^56$Ni mass, we show that R$_p$ <åisebox-0.5ex 0.02 R $_☉$ (a factor of five smaller than previously inferred), that the average density of the primary must be o̊$_ p $ > 10$^4$ g cm$^-3$, and that the effective temperature must be less than a few × 10$^5$ K. This rules out hydrogen-burning main-sequence stars and giants. Constructing the helium-burning and carbon-burning main sequences, we find that such objects are also excluded. By process of elimination, we find that only degeneracy-supported compact objects—WDs and neutron stars—are viable as the primary star of SN 2011fe. With few caveats, we also restrict the companion (secondary) star radius to R $_c$ <rs̊ebox-0.5ex 0.1 R $_☉$, excluding Roche-lobe overflowing red giant and main-sequence companions to high significance.