We use a sample of 45 low-metallicity H II regions to determine the primordial helium abundance Y-p with a precision of better than 1%. The data includes new spectrophotometric observations of 15 blue compact galaxies (BCGs) with oxygen abundance 12 + log (O/H) between 7.83 and 8.35 (Z(.)/13 less than or equal to Z less than or equal to Z(.)/4), most of which were selected from the First Byurakan and the University of Michigan objective prism surveys. We have included many low-metallicity BCGs in our sample, including the two most metal-deficient galaxies known, I ZW 18 (Z(.)/55) and SBS 0335-052 (Z(.)/43). We have carefully investigated the physical effects that may make the He I line intensities deviate from their recombination values, such as collisional and fluorescent enhancements, underlying He I stellar absorption, and absorption by Galactic interstellar Na I. By extrapolating the Y versus O/H and Y versus N/H linear regressions to O/H = N/H = 0, we obtain Y-p= 0.244 +/- 0.002 and 0.245 +/- 0.001, respectively, in agreement with the study of Izotov, Thuan, & Lipovetsky, but higher than previous determinations (Y-p = 0.230-0.234). Part of the difference comes from the fact that previous investigators have used the northwest component of I ZW 18 in the determination of Y-p. This component is subject to strong underlying He I stellar absorption that reduces the He I line intensities by 5%-25%. The derived Y is 0.233 +/- 0.008 from the He I lambda 6678 line. Instead, by using the southeast component of I ZW 18, which is much less subject to underlying He I stellar absorption, we obtain Y = 0.242 +/- 0.009. The mean Y of the two most metal-deficient BCGs, I ZW 18 and SBS 0335-052, is (Y) over bar = 0.245 +/- 0.004, in excellent agreement with the Y-p derived from the linear regressions. We derive a slope dY/dZ = 2.3 +/- 1.0, considerably smaller than those derived before. With this smaller slope and taking into account the errors, chemical evolution models with an outflow of well-mixed material can be built for star-forming dwarf galaxies that satisfy all the observational constraints. Our Y-p gives Omega(b) h(50)(2) = 0.058 +/- 0.007, consistent with the lower limit set by dynamical measurements and X-ray observations of clusters of galaxies. It is also consistent, within the framework of standard big bang nucleosynthesis theory, with measurements of primordial Li-7 in galactic halo stars, and at the 1 sigma level with the D/H abundance measured in absorption systems toward quasars by Tytler & Burles.
