  The aim of this paper is to introduce and to study the space
$\mathcal{O}_{M,\omega}(\mathbb{R}^N)$ of the multipliers of the space
$\mathcal{S}_{\omega}(\mathbb{R}^N)$ of the $\omega$-ultradifferentiable
rapidly decreasing functions of Beurling type. We determine various properties
of the space $\mathcal{O}_{M,\omega}(\mathbb{R}^N)$. Moreover, we define and
compare some lc-topologies of which $\mathcal{O}_{M,\omega}(\mathbb{R}^N)$ can
be naturally endowed.
