  We present an argument for a {\it lower limit} to the Compton-$y$ parameter
describing spectral distortions of the cosmic microwave background (CMB). The
absence of a detectable Gunn-Peterson signal in the spectra of high redshift
quasars demands a high ionization state of the intergalactic medium (IGM).
Given an ionizing flux at the lower end of the range indicated by the proximity
effect, an IGM representing a significant fraction of the
nucleosynthesis-predicted baryon density must be heated by sources other than
the photon flux to a temperature $\go {\rm few} \times 10^5\, K$. Such a gas at
the redshift of the highest observed quasars, $z\sim 5$, will produce a $y\go
10^{-6}$. This lower limit on $y$ rises if the Universe is open, if there is a
cosmological constant, or if one adopts an IGM with a density larger than the
prediction of standard Big Bang nucleosynthesis.
