  We present the results of a long-slit K-band spectroscopic search for H-alpha
emission from eight damped Ly-alpha absorbers (DLAs) at z>2 with the goal of
measuring the star-formation rates in these systems. For each system we
searched for compact sources of H-alpha emission within a solid angle
11x2.5arcsec^2 (44x10 kpc^2/h^2, for q_0=0.5, H_0=100 h km/s/Mpc). No H-alpha
emission was detected above 3-sigma limits in the range (6.5 - 16)x1E-20 W/m^2,
equivalent to star formation rates of (5.6 - 18) M_sun/yr/h^2, for a standard
IMF, assuming the lines are spectrally unresolved (<650 km/s FWHM). We compare
these results against the predictions of the models of Pei & Fall (1995) of the
global history of star formation, under two different simplifying hypotheses:
i.) the space density of DLAs at z=2.3 is equal to the space density of spiral
galaxies today (implying DLA disks were larger in the past, the `large-disk'
hypothesis); ii.) the sizes of DLAs at z=2.3 were the same as the gas sizes of
spiral galaxies today (implying DLA disks were more common in the past, the
`hierarchical' hypothesis). Compared to the previous most sensitive
spectroscopic search our sample is twice as large, our limits are a factor
greater than two deeper, and the solid angle surveyed is over three times as
great. Despite this our results are not in conflict with either the large-disk
hypothesis, because of the limited solid angle covered by the slit, or the
hierarchical hypothesis, because of the limited sensitivity.
