The authors are concerned with a critical (Bienaymé-)Galton-Watson process \((Z_n)_{n\geq 0}\) with \(Z_0=1, P(Z_1=k)=p_k, k\geq 0\), thus satisfying the condition \(\sum_{k\geq 1}kp_k=1\). It is also assumed that the variance \(B=\sum_{k\geq 2}k(k-1)p_k\) of \(Z_1\) is finite. Let \(d\) denote the greatest common divisor of the numbers \(k\geq 0\) for which \(p_k>0\). The authors prove that if both \(m\) and \(n\) tend to \(\infty\) such that the ratio \(m/n\) stays bounded, then \[ \lim_{n\to\infty}\frac{B^2n^2}{4d}\left(1+\frac{2d}{Bn}\right)^{m+1} P(Z_n=md)=1. \] The proof is very long and laborious (21 lemmas!). It is based on the fact that the assumptions made imply that the quantity \(\sup_{n,k\geq 1}n^2P(Z_n=k)\) is finite.