We have obtained deep Hubble Space Telescope far-UV images of 15 starburst galaxies at z~1.3 in the GOODS fields to search for escaping Lyman continuum photons. These are the deepest far-UV images m_{AB}=28.7, 3sigma, 1 diameter) over this large an area (4.83 arcmin^2) and provide the best escape fraction constraints for any galaxy at any redshift. We do not detect any individual galaxies, with 3sigma limits to the Lyman Continuum (~700 AA) flux 50--149 times fainter (in f_nu) than the rest-frame UV (1500 AA) continuum fluxes. Correcting for the mean IGM attenuation (factor ~2), as well as an intrinsic stellar Lyman Break (~3), these limits translate to relative escape fraction limits of f_{esc,rel}<[0.03,0.21]. The stacked limit is f_{esc,rel}(3sigma)<0.02. We use a Monte Carlo simulation to properly account for the expected distribution of IGM opacities. When including constraints from previous surveys at z~1.3 we find that, at the 95% confidence level, no more than 8% of star--forming galaxies at z~1.3 can have relative escape fractions greater than 0.50. Alternatively, if the majority of galaxies have low, but non-zero, escaping Lyman Continuum, the escape fraction can not be more than 0.04. Both the stacked limits, and the limits from the Monte Carlo simulation suggest that the average ionizing emissivity (relative to non-ionizing UV emissivity) at z~1.3 is significantly lower than has been observed in Lyman Break Galaxies (LBGs) at z~3. If the ionizing emissivity of star-forming galaxies is in fact increasing with redshift, it would help to explain the high photoionization rates seen in the IGM at z>4 and reionization of the intergalactic medium at z>6. [Abridged]
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