The Robotic Optical Transient Search Experiment (ROTSE) seeks to measure simultaneous and early afterglow optical emission from gamma-ray bursts (GRBs). A search for optical counterparts to six GRBs with localization errors of 1 deg2 or better produced no detections. The earliest limiting sensitivity is mROTSE>13.1 at 10.85 s (5 s exposure) after the gamma-ray rise, and the best limit is mROTSE>16.0 at 62 minutes (897 s exposure). These are the most stringent limits obtained for the GRB optical counterpart brightness in the first hour after the burst. Consideration of the gamma-ray fluence and peak flux for these bursts and for GRB 990123 indicates that there is not a strong positive correlation between optical flux and gamma-ray emission.