  Consider a free ergodic measure preserving profinite action
$\Gamma\curvearrowright X$ (i.e. an inverse limit of actions
$\Gamma\curvearrowright X_n$, with $X_n$ finite) of a countable property (T)
group $\Gamma$ (more generally of a group $\Gamma$ which admits an infinite
normal subgroup $\Gamma_0$ such that the inclusion $\Gamma_0\subset\Gamma$ has
relative property (T) and $\Gamma/\Gamma_0$ is finitely generated) on a
standard probability space $X$. We prove that if $w:\Gamma\times X\to \Lambda$
is a measurable cocycle with values in a countable group $\Lambda$, then $w$ is
cohomologous to a cocycle $w'$ which factors through the map $\Gamma\times X\to
\Gamma\times X_n$, for some $n$. As a corollary, we show that any orbit
equivalence of $\Gamma\curvearrowright X$ with any free ergodic measure
preserving action $\Lambda\curvearrowright Y$ comes from a (virtual) conjugacy
of actions.
