  Type Ia supernovae have proven vital to our understanding of cosmology, both
as standard candles and for their role in galactic chemical evolution; however,
their origin remains uncertain. The canonical accretion model implies a hot and
luminous progenitor which would ionize the surrounding gas out to a radius of
$\sim$10--100 parsecs for $\sim$100,000 years after the explosion. Here we
report stringent upper limits on the temperature and luminosity of the
progenitor of Tycho's supernova (SN 1572), determined using the remnant itself
as a probe of its environment. Hot, luminous progenitors that would have
produced a greater hydrogen ionization fraction than that measured at the
radius of the present remnant ($\sim$3 parsecs) can thus be excluded. This
conclusively rules out steadily nuclear-burning white dwarfs (supersoft X-ray
sources), as well as disk emission from a Chandrasekhar-mass white dwarf
accreting $\gtrsim 10^{-8}M_{\odot}$yr$^{-1}$ (recurrent novae). The lack of a
surrounding Str\"omgren sphere is consistent with the merger of a double white
dwarf binary, although other more exotic scenarios may be possible.
