  In this paper we address the questions of perfectly sampling a Gibbs measure
with infinite range interactions and of perfectly sampling the measure together
with its finite range approximations. We solve these questions by introducing a
perfect simulation algorithm for the measure and for the coupled measures. The
algorithm works for general Gibbsian interaction under requirements on the
tails of the interaction. As a consequence we obtain an upper bound for the
error we make when sampling from a finite range approximation instead of the
true infinite range measure.
