  We investigate the heat source of the neutral gas comprising DLAs. Unlike the
Lya forest, where the extragalactic background radiation field ionizes and
heats the gas, we find that grain photoelectric heating by the FUV background
is not sufficient to balance the C II 158um cooling rate inferred from DLAs. In
these systems, a local energy source is required. We show that in the case of
the z=1.919 DLA toward Q2206-19, the local source is FUV emission from the
associated galaxy found by Moller et al (2002): the mean intensity inferred
from photometry is in good agreement with the intensity Jnu required to explain
the cooling rate. The FUV mean intensity predicted for a cold neutral medium
(CNM) model, Jnu=(1.7+2.7-1.0)x10^(-18) cgs (95% c.l.), is the largest expected
from our CII* study of 45 DLAs. This may explain why this is the only confirmed
DLA yet detected in emission at z>1.9. We argue that in most DLAs with detected
CII* absorption, Jnu is between 10^{-19} and 10^{-18} and heats the gas which
is a CNM. By contrast, in most DLAs with upper limits on CII* absorption the
gas is a warm neutral medium (WNM). Surprisingly, the upper limits are
compatible with the same range of Jnu values suggesting the majority of DLAs
are heated by radiation fields generated by a limited range of star formation
rates per unit H I area, between 10^{-3} and 10^{-2} Msol/kpc^2. We also show
that CII* absorption is unlikely to arise in gas that is ionized.
