The mention of the weak force is something of a red herring as any interaction that allows protons and neutrons to interconvert would give the same result.
Earlier in the chapter the authors describe how to calculate the relative number densities of two components that are in equilibrium. For neutrons and protons they assume that these can be treated as a gas of free neutrons and free protons, and they derive the equation 4.17:
$$ \frac{n_p^{(0)}}{n_n^{(0)}} = \frac{ e^{-m_p/T} \int dp ~ p^2 e^{-p^2/2m_pT}}{e^{-m_n/T} \int dp ~ p^2 e^{-p^2/2m_nT}} \tag{4.17} $$
And since the masses of the neutron and proton are very similar the integrals are very similar and can be set equal to a good approximation. Then we get equation 4.18 for the proton-neutron ratio:
$$ \frac{n_p^{(0)}}{n_n^{(0)}} \approx \exp\left(\frac{m_n - m_p}{T}\right) \tag{4.18}$$
Since $m_n - m_p > 0$ the ratio $n_p/n_n \to \infty$ as $T \to 0$ i.e. all the neutrons convert into protons. This result depends only on the fact the neutron is more massive that the proton and it applies as long as the two particles can freely interconvert.
The weak interaction is mentioned because it is the means by which neutrons and proton can interconvert. There is nothing special about the weak interaction in this context as any interaction that allowed the interconversion would have the same result.
