The Robotic Optical Transient Search Experiment (ROTSE) seeks to measure si
multaneous and early afterglow optical emission from gamma-ray bursts (GRBs
). A search for optical counterparts to six GRBs with localization errors o
f 1 deg(2) or better produced no detections. The earliest limiting sensitiv
ity is m(ROTSE) > 13.1 at 10.85 s (5 s exposure) after the gamma-ray rise,
and the best limit is m(ROTSE) > 16.0 at 62 minutes (897 s exposure). These
are the most stringent limits obtained for the GRB optical counterpart bri
ghtness in the first hour after the burst. Consideration of the gamma-ray f
luence and peak flux for these bursts and for GRB 990123 indicates that the
re is not a strong positive correlation between optical flux and gamma-ray
emission.