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We present UV measurements of O~III] 1666~\AA\ and C~III] 1909~\AA\ emission lines in giant H~II regions in low-luminosity dwarf irregular galaxies and the Magellanic Clouds obtained with the FOS on HST in order to measure the C/O abundance ratio in the ISM of those galaxies. We find a continuous increase in C/O with increasing O/H, consistent with a power law having an index of 0.43$\pm$0.09 over the range $-$4.7 to $-$3.6 in log(O/H). One possible interpretation of this trend is that the most metal-poor galaxies are the youngest and dominated by the products of early enrichment by massive stars, while more metal-rich galaxies show increasing, delayed contributions of carbon from intermediate mass stars. However, recent evolution models for massive stars including mass loss suggest that the yield of carbon from massive stars may increase with metallicity relative to the yield of oxygen; new chemical evolution models for the solar neighborhood which include nucleosynthesis from these recent stellar evolution models predict a C/O abundance evolution similar to that observed in the metal-poor galaxies. The trend in the C/N ratio increases steadily with O/H in the irregular galaxies, but C/N is significantly smaller for solar neighborhood stars and H II regions. This may indicate that the bulk of nitrogen production is decoupled from the synthesis of carbon in our Galaxy, and that the star formation history in irregulars is very different from that for the Galaxy. Our measured C/O ratios in the most metal-poor galaxies are consistent with predictions of nucleosynthesis from massive stars for Weaver and Woosley's best estimate for the $^{12}C(\alpha,\gamma)^{16}O$ nuclear reaction rate, assuming negligible contamination from carbon produced in intermediate mass stars in these galaxies.
