  We are interested in the Euler-Maruyama discretization of a stochastic
differential equation in dimension $d$ with constant diffusion coefficient and
bounded measurable drift coefficient. In the scheme, a randomization of the
time variable is used to get rid of any regularity assumption of the drift in
this variable. We prove weak convergence with order $1/2$ in total variation
distance. When the drift has a spatial divergence in the sense of distributions
with $\rho$-th power integrable with respect to the Lebesgue measure in space
uniformly in time for some $\rho \ge d$, the order of convergence at the
terminal time improves to $1$ up to some logarithmic factor. In dimension
$d=1$, this result is preserved when the spatial derivative of the drift is a
measure in space with total mass bounded uniformly in time. We confirm our
theoretical analysis by numerical experiments.
