  In this paper, we study the asymptotic behaviour of one-dimensional
integrated Ornstein-Uhlenbeck processes driven by $\alpha$-stable L\'{e}vy
processes of small amplitude. We prove that the integrated Ornstein-Uhlenbeck
process converges weakly to the underlying $\alpha$-stable L\'{e}vy process in
the Skorokhod $M_1$-topology which secures the weak convergence of first
passage times. This result follows from a more general result about
approximations of an arbitrary L\'{e}vy process by continuous integrated
Ornstein-Uhlenbeck processes in the $M_1$-topology.
