  We show that the epoch(s) of reionization when the ionization fraction of the
universe is about half can be determined by correlating Cosmic Microwave
Background (CMB) temperature maps with 21-cm line maps at degree scales ($l\sim
100$). During reionization peculiar motion of free electrons induces the
Doppler anisotropy of the CMB, while density fluctuations of neutral hydrogen
induce the 21-cm line anisotropy. In our simplified model of inhomogeneous
reionization, a positive correlation arises as the universe reionizes whereas a
negative correlation arises as it recombines; thus, the sign of the correlation
provides information on the reionization history which cannot be obtained by
presently. The signal comes mainly from large scales (k~0.01 Mpc^-1) where
linear perturbation theory is valid and complexity due to patchy reionization
is averaged out. Since the Doppler signal comes from ionized regions and the
21-cm comes from neutral ones, the correlation has a well-defined peak(s) in
redshift when the ionization fraction of the universe is about half.
Furthermore, the cross-correlation is much less sensitive to systematic errors,
especially foreground emission, than the auto-correlation of 21-cm lines: this
is analogous to the temperature-polarization correlation of the CMB being more
immune to systematic errors than the polarization-polarization. Therefore, we
argue that the Doppler-21cm correlation provides a robust measurement of the
21-cm anisotropy, which can also be used as a diagnostic tool for detected
signals in the 21-cm data -- detection of the cross-correlation provides the
strongest confirmation that the signal is of cosmological origin. We show that
the Square Kilometer Array can easily measure the predicted correlation signal
for 1~year of survey observation.
