For $n = 1,2, \cdots$ let $z^{(n)}_t, t \geqq 0$, be an age-dependent branching process starting from $n$ ancestors. Suppose it has the reproduction generating function $f_n, f_n'(1) = 1 + \alpha/n + o(n^{-1}), f_n''(1) = 2\beta_n \rightarrow 2\beta, f_n'''(1-) \leqq$ some constant, and the life-length distribution $L$ with $L(0) = 0$ and $\lambda = \int^\infty_0 tL(dt) < \infty$. Then, it is shown that the finite dimensional distributions of $n^{-1}z^{(n)}_{nt}$ converge, as $n \rightarrow \infty$, to the corresponding laws of the diffusion $t \rightarrow x_t$ with drift $(\alpha/\lambda)x$ and infinitesimal variance $(2\beta/\lambda)x$.