We use spectroscopic observations of a sample of 82 H II regions in 76 blue compact galaxies to determine the primordial helium abundance Y_p and the slope dY/dZ from the Y-O/H linear regression. To improve the accuracy of the dY/dZ measurement, we have included new spectrophotometric observations of 33 H II regions that span a large metallicity range, with oxygen abundance 12 + log(O/H) varying between 7.43 and 8.30 (Z_☉/30 ≤ Z ≤ Z_☉/4). Most of the new galaxies were selected from the First Byurakan, the Hamburg/SAO, and the University of Michigan objective prism surveys. For a subsample of seven H II regions, we derive the He mass fraction taking into account known systematic effects, including collisional and fluorescent enhancements of He I emission lines, collisional excitation of hydrogen emission, underlying stellar He I absorption, and the difference between the temperatures T_e(He II) in the He⁺ zone and T_e(O III) derived from the collisionally excited [O III] lines. We find that the net result of all the systematic effects combined is small, changing the He mass fraction by less than 0.6%. By extrapolating the Y versus O/H linear regression to O/H = 0 for seven H II regions of this subsample, we obtain Y_p = 0.2421 ± 0.0021 and dY/dO = 5.7 ± 1.8, which corresponds to dY/dZ = 3.7 ± 1.2, assuming the oxygen mass fraction to be O = 0.66Z. In the framework of the standard big bang nucleosynthesis theory, this Y_p corresponds to Ω_bh² = 0.012, where h is the Hubble constant in units of 100 km s^-1 Mpc^-1. This is smaller at the 2 σ level than the value obtained from recent deuterium abundance and microwave background radiation measurements. The linear regression slope dY/dO = 4.3 ± 0.7 (corresponding to dY/dZ = 2.8 ± 0.5) for the whole sample of 82 H II regions is similar to that derived for the subsample of seven H II regions, although it has a considerably smaller uncertainty.