  Consider $n \geq 2$. In this paper we prove that the group $\text{PU}(n,1)$
is $1$-taut. This result concludes the study of $1$-tautness of rank-one Lie
groups of non-compact type. Additionally the tautness property implies a
classification of finitely generated groups which are $\text{L}^1$-measure
equivalent to lattices of $\text{PU}(n,1)$. More precisely, we show that
$\text{L}^1$-measure equivalent groups must be extensions of lattices of
$\text{PU}(n,1)$ by a finite group.
