Let $L= -\Delta+ V$ be a Schrödinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative function, $V\ne 0$, and belongs to the reverse Hölder class $RH_{d/2}$. In this paper, we prove a version of the classical theorem of Jones and Journé on weak$^*$-convergence in the Hardy space $H^1_L(\mathbb R^d)$.