  In this paper, we are interested in the problem of scattering by strictly
convex obstacles in the plane. We provide an upper bound for the number
$N(r,\gamma)$ of resonances in the box $\{r \le \Re(\lambda) \le r + 1$;
$\Im(\lambda) \ge - \gamma \}$. It was proved in a work of S. Nonnenmacher, J.
Sj{\"o}strand and M. Zworski (2014) that $N (r,\gamma) = O_\gamma (r^{d_H})$
where $2d_H + 1$ is the Hausdorff dimension of the trapped set of the billiard
flow. In this article, we provide an improved upper bound in the band $0 \le
\gamma < \gamma_{cl} /2$, where $\gamma_{cl}$ is the classical decay rate of
the flow. This improved Weyl upper bound is in the spirit of the ones of F.
Naud (2012) and S. Dyatlov (2019) in the case of convex co-compact surfaces,
and of S. Dyatlov and L. Jin (2017) in the case of open quantum baker's maps.
