A limit theorem for the largest interpoint distance of $p$ i.i.d. points on $\mathbb R^n$ to the Gumbel distribution is proven, where the number of points $p=p_n$ tends to infinity as the dimension of the points $n$ tends to infinity. The theorem holds under moment assumptions and corresponding assumptions on the rate of $p$. The proof is based on the Chen-Stein Poisson approximation method and uses the sum structure of the interpoint distances. Therefore, an asymptotic distribution of a more general object is derived.