  We present the first large, blind and uniformly selected survey for molecular
hydrogen (H_2) in damped Lyman alpha systems (DLAs) with moderate-to-high
resolution spectra. 86 DLAs were searched for absorption in the many Lyman and
Werner H_2 transitions, with approximately 79% completeness for H_2 column
densities above N(H_2)=10^17.5 cm^-2 for an assumed Doppler broadening
parameter b=2 km s^-1. Only a single strong H_2 absorber was found -- a system
detected previously in VLT/UVES spectra. Given our distribution of N(H_2) upper
limits, this ~1% detection rate is smaller than expected from previous surveys
at 99.8% confidence. Assuming the N(H_2) distribution shape from previous
surveys, our detection rate implies a covering factor of ~1% for N(H_2) >=
10^17.5 cm^-2 gas in DLAs (<6% at 95% confidence). We obtained new
Magellan/MagE spectra for 53 DLAs; 8 km s^-1 -resolution spectra were available
for 27 DLAs. MagE's moderate resolution (~71 km s^-1) yields weaker N(H_2)
upper limits and makes them dependent on the assumed Doppler parameter. For
example, half the (relevant) previous H_2 detections have N(H_2) >= 10^18.1
cm^-2, a factor of just 3 higher than our median upper limit. Nevertheless,
several tests suggest our upper limits are accurate, and they would need to be
increased by 1.8 dex to bring our detection rate within 95% confidence of
previous surveys.
