  We prove a Berry-Esseen bound in de Jong's classical CLT for normalized,
completely degenerate $U$-statistics, which says that the convergence of the
fourth moment sequence to three and a Lindeberg-Feller type negligibility
condition are sufficient for asymptotic normality. Our bound is of the same
optimal order as the bound on the Wasserstein distance to normality that has
recently been proved by D\"obler and Peccati (2017).
