We present near-infrared spectroscopic observations of the high-intensity He i λ10830Å emission line in 45 low-metallicity H ii regions. We combined these NIR data with spectroscopic data in the optical range to derive the primordial He abundance. The use of the He i λ10830Å line, the intensity of which is very sensitive to the density of the H ii region, greatly improves the determination of the physical conditions in the He + zone. This results in a considerably tighter Y -O/H linear regression compared to all previous studies. We extracted a final sample of 28 H ii regions with Hβ equivalent width EW(Hβ) 150Å, excitation parameter O 2+ /O 0.8, and with helium mass fraction Y derived with an accuracy better than 3%. With this final sample we derived a primordial 4 He mass fraction Y p = 0.2551 ± 0.0022. The derived value of Y p is higher than the one predicted by the standard big bang nucleosynthesis (SBBN) model. Using our derived Y p together with D/H = (2.53±0.04)×10 −5 , and the χ 2 technique, we found that the best agreement between these light element abundances is achieved in a cosmological model with a baryon mass density Ω b h 2 = 0.0240±0.0017 (68% CL), ±0.0028 (95.4% CL), ±0.0034 (99% CL) and an effective number of neutrino species N eff = 3.58±0.25 (68% CL), ±0.40 (95.4% CL), ±0.50 (99% CL). A non-standard value of N eff is preferred at the 99% CL, implying the possible existence of additional types of neutrino species.