  We present a new sample of 35 quasars selected from the FIRST radio survey
and the Deeprange I-band survey (Postman et al. 1998, 2002). A comparison with
the FIRST Bright Quasar survey samples reveals that this I-band selected sample
is redder by 0.25-0.5 mag in B-R, and that the color difference is not
explained by the higher mean redshift of this sample but must be intrinsic. Our
small sample contains five quasars with unusually red colors, including three
that appear very heavily reddened. Our data are fitted well with normal blue
quasar spectra attenuated by more than 2.5 magnitudes of extinction in the
I-band.
  The red quasars are only seen at low redshifts (z<1.3). Even with a magnitude
limit I<20.5, our survey is deep enough to detect only the most luminous of
these red quasars at z~1; similar objects at higher redshifts would fall below
our in I-band limit. Indeed, the five most luminous objects (using dereddened
magnitudes) with z<1.3 are all red. Our data strongly support the hypothesis
that radio quasars are dominated by a previously undetected population of red,
heavily obscured objects. Unless highly reddened quasars are preferentially
also highly luminous, there must be an even larger, as yet undiscovered,
population of red quasars at lower luminosity. We are likely to be finding only
the most luminous tip of the red quasar iceberg.
  Five of our six z<1 quasars are associated with Deeprange cluster candidates
of similar estimated redshifts. This association is very unlikely to be the
result of chance. It has some surprising implications, including the
possibility that up to half of the Deeprange clusters at z~1 have associated
quasars.
