  We describe a technique that for the first time measures star formation rates
(SFRs) in damped Lyman alpha systems(DLAs) directly. We assume that massive
stars form in DLAs, and that the FUV radiation they emit heats the gas by the
grain photoelectric mechanism. We infer the heating rate from the cooling rate
measured by the strength of CII* 1335.7 absorption. Since the heating rate is
proportional to the dust-to-gas ratio and the SFR per unit area, we deduce the
SFR per unit area for DLAs in which both quantities have been measured. We
consider models in which the the dust comprises carbonaceous or silicate
grains. We present two-phase models where the cold neutral medium (CNM) and
warm neutral medium (WNM) are in pressuer equilibrium. In the CNM model the the
sightline goes throught the CNM and WNM, while in the WNM model it goes only
through the WNM. Since the grain photoelectric heating efficiency is at least
10 times higher in the CNM than in the WNM, CII* absorption mainly arises in
the CNM in the CNM model. But in the WNM model all of the CII* absorption
arises in the WNM. We use CII* absorption lines to derive the SFR per unit area
for a sample of ~ 30 DLAs in which the dust-to-gas ratio has been inferred from
element depletion patterns. We show that the resulting SFR per unit area
corresponds to an average over the star forming volume of galaxy hosting the
DLA rather than to local star formation along the line of sight. We find the
average SFR per unit area and equals10$^{-2.2}$ M$_{\odot}$yr$^{-1}$kpc$^{-2}$
for the CNM model and 10$^{-1.3}$ M$_{\odot}$yr$^{-1}$kpc$^{-2}$ for the WNM
model. The SFR per unit area in the CNM solution is similar to that measured in
the Milky Way ISM.
