  The classical Berry-Esseen error bound, for the normal approximation to the
law of a sum of independent and identically distributed random variables, is
here improved by replacing the standardised third absolute moment by a weak
norm distance to normality. We thus sharpen and simplify two results of Ulyanov
(1976) and of Senatov (1998), each of them previously optimal, in the line of
research initiated by Zolotarev (1965) and Paulauskas (1969).
  Our proof is based on a seemingly incomparable normal approximation theorem
of Zolotarev (1986), combined with our main technical result:
  The Kolmogorov distance (supremum norm of difference of distribution
functions) between a convolution of two laws and a convolution of two Lipschitz
laws is bounded homogeneously of degree 1 in the pair of the Wasserstein
distances (L$^1$ norms of differences of distribution functions) of the
corresponding factors, and also, inessentially for the present application, in
the pair of the Lipschitz constants.
  Side results include a short introduction to $\zeta$ norms on the real line,
simpler inequalities for various probability distances, slight improvements of
the theorem of Zolotarev (1986) and of a lower bound theorem of Bobkov,
Chistyakov and G\"otze (2012), an application to sampling from finite
populations, auxiliary results on rounding and on winsorisation, and
computations of a few examples.
  The introductory section in particular is aimed at analysts in general rather
than specialists in probability approximations.
