We present the results of rest-frame, UV slitless spectroscopic observations of a sample of 32 z~0.7 Lyman break galaxy (LBG) analogs in the COSMOS field. The spectroscopic search was performed with the Solar Blind Channel (SBC) on Hubble Space Telescope. We report the detection of leaking Lyman continuum (LyC) radiation from an AGN-starburst composite. While we find no direct detections of LyC emission in the remainder of our sample, we achieve individual lower limits (3 sigma) of the observed non-ionizing UV to LyC flux density ratios, f_{nu}(1500A)/f_{nu}(830A) of 20 to 204 (median of 73.5) and 378.7 for the stack. Assuming an intrinsic Lyman break of 3.4 and an intergalactic medium (IGM) transmission of LyC photons along the line of sight to the galaxy of 85% we report an upper limit for the relative escape fraction in individual galaxies of 0.02 - 0.19 and a stacked 3 sigma upper limit of 0.01. We find no indication of a relative escape fraction near unity as seen in some LBGs at z~3. Our UV spectra achieve the deepest limits to date at any redshift for the escape fraction in individual sources. The contrast between these z~0.7 low escape fraction LBG analogs with z~3 LBGs suggests that either the processes conducive to high escape fractions are not being selected for in the z<1 samples or the average escape fraction is decreasing from z~3 to z~1. We discuss possible mechanisms which could affect the escape of LyC photons.