We study the effect that the temperature structure has on the determination of the primordial helium abundance, Yp. We provide an equation linking T(O III), the temperature derived from the [O III] lines, and T(He II), the temperature of the He I lines, both for H II regions with O++ only and for H II regions where a fraction of O+ is present. By means of T(He II), which is always smaller than T(O III), we derive the helium abundances of five objects with low and very low metallicity (NGC 346, NGC 2363, Haro 29, SBS 0335-052, and I Zw 18); these objects were selected from the literature because they include the three low-metallicity objects with the best line determinations and the two objects with the lowest metallicity. From these abundances we obtain that Yp(nHc)=0.2356+/-0.0020, a value 0.0088 lower than that derived by using T(O III). We call this determination Yp(nHc) because the collisional contribution to the Balmer-line intensities has not been taken into account. All the recent Yp determinations in the literature have not taken into account the collisional contribution to the Balmer-line intensities. By considering the collisional contribution to the Balmer-line intensities of these five objects, we derive that Yp(+Hc)=0.2384+/-0.0025.