We determine, asymptotically in $n$, the distribution and mean of the weight of a minimum-weight $k$-clique (or any strictly balanced graph $H$) in a complete graph $K_n$ whose edge weights are independent random values drawn from the uniform distribution or other continuous distributions. For the clique, we also provide explicit (non-asymptotic) bounds on the distribution's CDF in a form obtained directly from the Stein-Chen method, and in a looser but simpler form. The direct form extends to other subgraphs and other edge-weight distributions. We illustrate the clique results for various values of $k$ and $n$. The results may be applied to evaluate whether an observed minimum-weight copy of a graph $H$ in a network provides statistical evidence that the network's edge weights are not independently distributed but have some structure.