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We use a sample of 45 low-metallicity H II regions to determine the primordial helium abundance Yp 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 ≤ Z ≤ 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 Yp = 0.244 ± 0.002 and 0.245 ± 0.001, respectively, in agreement with the study of Izotov, Thuan, & Lipovetsky, but higher than previous determinations (Yp = 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 Yp. 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 λ6678 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 =0.245±0.004, in excellent agreement with the Yp 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 Yp gives Ωb h250=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 7Li in galactic halo stars, and at the 1 σ level with the D/H abundance measured in absorption systems toward quasars by Tytler & Burles.