  We consider the model of random interlacements on $\mathbb{Z}^d$ introduced
in Sznitman [Vacant set of random interlacements and percolation (2007)
preprint]. For this model, we prove the uniqueness of the infinite component of
the vacant set. As a consequence, we derive the continuity in $u$ of the
probability that the origin belongs to the infinite component of the vacant set
at level $u$ in the supercritical phase $u<u_*$.
