We investigate the weight distribution of random binary linear codes. For $0<λ<1$ and $n\to\infty$ pick uniformly at random $λn$ vectors in $\mathbb{F}_2^n$ and let $C \le \mathbb{F}_2^n$ be the orthogonal complement of their span. Given $0<γ<1/2$ with $0< λ< h(γ)$ let $X$ be the random variable that counts the number of words in $C$ of Hamming weight $γn$. In this paper we determine the asymptotics of the moments of $X$ of all orders $o(\frac{n}{\log n})$.