  We present predictions for the reionization of the intergalactic medium by
stars in high-redshift galaxies. We calculate ionizing luminosities of
galaxies, including the effects of absorption by interstellar gas and dust on
the escape fraction, in order to calculate the filling factor of ionized
hydrogen in the IGM. For a $\Lambda{\rm CDM}$ cosmology and a physical
calculation of the escape fraction, we find that the hydrogen in the IGM will
be reionized at redshift z=6.1 if the IGM has uniform density, but only by
z=4.5 if the IGM is clumped. We combine our model with an N-body simulation of
the distribution of dark matter in the universe in order to calculate the
secondary temperature anisotropies induced in the cosmic microwave background
(CMB) by scattering off free electrons. The models predict a spectrum of
secondary anisotropies covering a broad range of angular scales, with
fractional temperature fluctuations $\sim 10^{-7}-10^{-6}$ on arcminute scales.
The amplitude depends somewhat on the geometry of reionization, with models in
which the regions of highest gas density are reionized first giving larger CMB
fluctuations than the case where galaxies ionize surrounding spherical regions,
and models where low density regions reionize first giving the smallest
fluctuations. Measurement of these anisotropies can therefore put important
constraints on the reionization process, in particular, the redshift evolution
of the filling factor, and should be a primary objective of a next generation
submillimeter telescope such as the Atacama Large Millimeter Array. (abrdiged)
