  In this paper we introduce a new atomic Hardy space $X^1(\gamma)$ adapted to
the Gauss measure $\gamma$, and prove the boundedness of the first order Riesz
transform associated with the Ornstein-Uhlenbeck operator from $X^1(\gamma)$ to
$L^1(\gamma)$. We also provide a new, short and almost self-contained proof of
its weak-type $(1,1)$.
